Optimal. Leaf size=11 \[ -\log (x)+\log \left (1+x^3\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 78}
\begin {gather*} \log \left (x^3+1\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{3} \text {Subst}\left (\int \frac {-1+2 x}{x (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {3}{1+x}\right ) \, dx,x,x^3\right )\\ &=-\log (x)+\log \left (1+x^3\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\log (x)+\log \left (1+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 19, normalized size = 1.73
method | result | size |
meijerg | \(-\ln \left (x \right )+\ln \left (x^{3}+1\right )\) | \(12\) |
risch | \(-\ln \left (x \right )+\ln \left (x^{3}+1\right )\) | \(12\) |
default | \(\ln \left (x +1\right )-\ln \left (x \right )+\ln \left (x^{2}-x +1\right )\) | \(19\) |
norman | \(\ln \left (x +1\right )-\ln \left (x \right )+\ln \left (x^{2}-x +1\right )\) | \(19\) |
parallelrisch | \(\ln \left (x +1\right )-\ln \left (x \right )+\ln \left (x^{2}-x +1\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 13, normalized size = 1.18 \begin {gather*} \log \left (x^{3} + 1\right ) - \frac {1}{3} \, \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 11, normalized size = 1.00 \begin {gather*} \log \left (x^{3} + 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.73 \begin {gather*} - \log {\left (x \right )} + \log {\left (x^{3} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 13, normalized size = 1.18 \begin {gather*} \log \left ({\left | x^{3} + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 11, normalized size = 1.00 \begin {gather*} \ln \left (x^3+1\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 10, normalized size = 0.91 \begin {gather*} \ln \left (\frac {x^{3}+1}{x^{3}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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