Optimal. Leaf size=8 \[ -\log (1-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1600, 31}
\begin {gather*} -\log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 1600
Rubi steps
\begin {align*} \int \frac {1+x+x^2+x^3+x^4}{1-x^5} \, dx &=\int \frac {1}{1-x} \, dx\\ &=-\log (1-x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -\log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 7, normalized size = 0.88
method | result | size |
default | \(-\ln \left (x -1\right )\) | \(7\) |
norman | \(-\ln \left (x -1\right )\) | \(7\) |
risch | \(-\ln \left (x -1\right )\) | \(7\) |
meijerg | \(-\frac {\ln \left (-x^{5}+1\right )}{5}-\frac {x^{4} \left (\ln \left (1-\left (x^{5}\right )^{\frac {1}{5}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )+2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )+2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )\right )}{5 \left (x^{5}\right )^{\frac {4}{5}}}-\frac {x^{3} \left (\ln \left (1-\left (x^{5}\right )^{\frac {1}{5}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )+2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )\right )}{5 \left (x^{5}\right )^{\frac {3}{5}}}-\frac {x^{2} \left (\ln \left (1-\left (x^{5}\right )^{\frac {1}{5}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )+2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )\right )}{5 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {x \left (\ln \left (1-\left (x^{5}\right )^{\frac {1}{5}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}+\left (x^{5}\right )^{\frac {2}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{5}\right )^{\frac {1}{5}}}\right )\right )}{5 \left (x^{5}\right )^{\frac {1}{5}}}\) | \(542\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 6, normalized size = 0.75 \begin {gather*} -\log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 6, normalized size = 0.75 \begin {gather*} -\log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 5, normalized size = 0.62 \begin {gather*} - \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 7, normalized size = 0.88 \begin {gather*} -\log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 6, normalized size = 0.75 \begin {gather*} -\ln \left (x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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