Optimal. Leaf size=18 \[ -\frac {1}{2 x^2}-\log (x)+\log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 1266,
908} \begin {gather*} -\frac {1}{2 x^2}+\log \left (x^2+1\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 1266
Rule 1607
Rubi steps
\begin {align*} \int \frac {1+x^4}{x^3+x^5} \, dx &=\int \frac {1+x^4}{x^3 \left (1+x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x^2 (1+x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {2}{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-\log (x)+\log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{2 x^2}-\log (x)+\log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 17, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{2 x^{2}}-\ln \left (x \right )+\ln \left (x^{2}+1\right )\) | \(17\) |
norman | \(-\frac {1}{2 x^{2}}-\ln \left (x \right )+\ln \left (x^{2}+1\right )\) | \(17\) |
meijerg | \(-\frac {1}{2 x^{2}}-\ln \left (x \right )+\ln \left (x^{2}+1\right )\) | \(17\) |
risch | \(-\frac {1}{2 x^{2}}-\ln \left (x \right )+\ln \left (x^{2}+1\right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{2 \, x^{2}} + \log \left (x^{2} + 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 25, normalized size = 1.39 \begin {gather*} \frac {2 \, x^{2} \log \left (x^{2} + 1\right ) - 2 \, x^{2} \log \left (x\right ) - 1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.83 \begin {gather*} - \log {\left (x \right )} + \log {\left (x^{2} + 1 \right )} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.48, size = 23, normalized size = 1.28 \begin {gather*} \frac {x^{2} - 1}{2 \, x^{2}} + \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 16, normalized size = 0.89 \begin {gather*} \ln \left (x^2+1\right )-\ln \left (x\right )-\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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