Optimal. Leaf size=18 \[ -\frac {1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1802, 635, 203, 260} \[ -\frac {1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1802
Rubi steps
\begin {align*} \int \frac {1-x^3}{x \left (1+x^2\right )} \, dx &=\int \left (-1+\frac {1}{x}+\frac {1-x}{1+x^2}\right ) \, dx\\ &=-x+\log (x)+\int \frac {1-x}{1+x^2} \, dx\\ &=-x+\log (x)+\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=-x+\tan ^{-1}(x)+\log (x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {1}{2} \log \left (x^2+1\right )-x+\log (x)+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 16, normalized size = 0.89 \[ -x + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 17, normalized size = 0.94 \[ -x + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 17, normalized size = 0.94 \[ -x +\arctan \relax (x )+\ln \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 16, normalized size = 0.89 \[ -x + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 1.33 \[ \ln \relax (x)-x+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 15, normalized size = 0.83 \[ - x + \log {\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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