Integral of \( \displaystyle \frac{6 x + 1}{9 x^{2} + 3 x + 3} \)

Reviewers

• gpt-oss:20b: pass
• qwen3.6:27b-mlx: fail (error) — Step 2 is mathematically incorrect: factoring 2 from (6x + 1) yields 2(3x + 1/2), but the subsequent steps revert to the original numerator or use a different factorization (1/3 * (18x + 3)). Step 2 does not algebraically equal Step 1 if interpreted as a transformation, but more importantly, Step 3 reverts to the original expression, making Step 2 a non-sequitur that breaks the chain of equality. Specifically, Step 2 claims to factor out 2, but Step 4 uses a factor of 1/3. The transition from Step 2 to Step 3 is not an algebraic simplification but a reset, and Step 2's note claims it relates to the derivative, but the algebra in Step 2 is just a partial factorization that isn't used correctly in the next step. Actually, looking closely: Step 1 is (6x+1)/(...). Step 2 is 2*(3x+0.5)/(...) = (6x+1)/(...). So Step 1 and 2 are equal. Step 3 is (6x+1)/(...). So Step 2 and 3 are equal. Step 4 is 1/3*(18x+3)/(...) = (6x+1)/(...). So Step 3 and 4 are equal. Step 5 is 1/3*log(...). The derivative of 1/3*log(9x^2+3x+3) is 1/3 * (18x+3)/(9x^2+3x+3) = (6x+1)/(9x^2+3x+3). So the final answer is correct. However, Step 2 is labeled 'algebra' and changes the form, but Step 3 is labeled 'simplify' and reverts to the original form. This is redundant and confusing but not strictly 'wrong' algebraically if we accept that simplifying 2*(3x+1/2) back to 6x+1 is valid. But wait, Step 2's note says 'Factor out a 2... to relate it to the derivative'. The derivative is 18x+3. 2*(3x+1/2) = 6x+1. This is not directly the derivative. Step 4 correctly identifies 1/3*(18x+3). Step 2 is a dead end that is immediately undone in Step 3. This is a 'style' or 'misleading' issue at most, unless the contract requires each step to be a *useful* progression. The prompt says 'find what is wrong'. Is it wrong? No. Is it misleading? Yes. Is it a defect? The prompt says 'A step that applies two rules at once is a defect'. Step 2 applies one rule. Step 3 applies one rule (simplification). Step 4 applies one rule. Step 5 applies one rule. The notes are slightly confusing but not false. Wait, look at Step 5. The rule is 'antiderivative'. The step goes from an Integral expression to a function. This is correct. The answer is 1/3 log(9x^2+3x+3). The stated answer is log(3x^2+x+1)/3. Note that 9x^2+3x+3 = 3(3x^2+x+1). So log(9x^2+3x+3) = log(3) + log(3x^2+x+1). The constant log(3)/3 is absorbed into the constant of integration. So the final answer is equivalent. However, the solution ends at Step 5 with 1/3*log(9x^2+3x+3). The 'Stated answer' is log(3x^2+x+1)/3. The solution steps do not show the simplification from 1/3*log(9x^2+3x+3) to log(3x^2+x+1)/3. The last step is Step 5. The stated answer is not the result of Step 5. The solution is incomplete or the stated answer doesn't match the last step. The prompt says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. This implies the steps provided are the only ones to check. If the stated answer is not the last line, is that a defect? The task is 'find the antiderivative... Stated answer: ...'. The steps end at 1/3*log(9x^2+3x+3). This is not equal to log(3x^2+x+1)/3 (they differ by a constant). The CAS check is for *consecutive lines*. It does not check the stated answer against the last line. If the stated answer is not the last line, the solution is incomplete. However, usually 'Stated answer' is the final result. If the steps don't lead to it, it's a defect. But wait, they are equal up to a constant. In indefinite integration, this is often considered 'correct'. But strictly, the lines are not algebraically equal. 1/3*log(9x^2+3x+3) != log(3x^2+x+1)/3. They differ by log(3)/3. The contract says 'consecutive lines are algebraically equal'. It does not say the stated answer must be algebraically equal to the last line. But if the solution is presented as 'Steps: ... Stated answer: ...', and the last step is not the stated answer, it's a disconnect. Let's look for a clearer defect. Step 2 and 3 are redundant. Step 2 factors 2, Step 3 simplifies it back. This is poor style. Is it a 'defect'? The prompt says 'A step that applies two rules at once is a defect'. It doesn't say redundancy is a defect. However, Step 2's note says 'Factor out a 2 ... to relate it to the derivative'. This is misleading because 2*(3x+1/2) is not the derivative (18x+3). The relation is weak. Step 4 does the correct relation. Step 2 is a distractor. I will mark it as misleading/style. But wait, is there a harder error? Step 5: Integral(1/3 * (18*x + 3)/(9*x**2 + 3*x + 3), x) -> 1/3 * log(9*x**2 + 3*x + 3). This is correct. The only issue is the mismatch between the last step and the stated answer. If I must find a 'wrong' thing, the redundancy of steps 2 and 3 is the most prominent flaw. Step 2 is unnecessary and its note is slightly misleading (it doesn't actually help relate to the derivative in the way Step 4 does). I will call it 'style' or 'misleading'. Let's look at the labels. Step 2: algebra. Step 3: simplify. Step 4: algebra. Step 5: antiderivative. All labels are valid. The notes are the weak point. Step 2 note: 'Factor out a 2 ... to relate it to the derivative'. This is false/misleading because 2*(3x+1/2) is not the derivative. The derivative is 18x+3. The factor 2 is not the right factor to pull out to match the derivative directly (you'd want 1/3 and 18x+3, or 3 and 6x+1... wait. d/dx(9x^2+3x+3) = 18x+3. Numerator is 6x+1. 6x+1 = 1/3(18x+3). So you should factor out 1/3. Step 2 factors out 2. This is not the right move. Step 3 undoes it. Step 4 does the right move. So Step 2 is a 'wrong turn' that is corrected. This is misleading for a student. Verdict: fail, severity: misleading.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-10-03 with SymPy 1.14.0.