Optimal. Leaf size=24 \[ -\frac {1}{x}+\tan ^{-1}(x)+2 \log (1-x)-\log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6857, 649, 209,
266} \begin {gather*} \text {ArcTan}(x)-\log \left (x^2+1\right )-\frac {1}{x}+2 \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 6857
Rubi steps
\begin {align*} \int \frac {-1+x+4 x^3}{(-1+x) x^2 \left (1+x^2\right )} \, dx &=\int \left (\frac {2}{-1+x}+\frac {1}{x^2}+\frac {1-2 x}{1+x^2}\right ) \, dx\\ &=-\frac {1}{x}+2 \log (1-x)+\int \frac {1-2 x}{1+x^2} \, dx\\ &=-\frac {1}{x}+2 \log (1-x)-2 \int \frac {x}{1+x^2} \, dx+\int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{x}+\tan ^{-1}(x)+2 \log (1-x)-\log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -\frac {1}{x}+\tan ^{-1}(x)+2 \log (1-x)-\log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 23, normalized size = 0.96
method | result | size |
default | \(-\frac {1}{x}+2 \ln \left (-1+x \right )-\ln \left (x^{2}+1\right )+\arctan \left (x \right )\) | \(23\) |
risch | \(-\frac {1}{x}+2 \ln \left (-1+x \right )-\ln \left (x^{2}+1\right )+\arctan \left (x \right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 22, normalized size = 0.92 \begin {gather*} -\frac {1}{x} + \arctan \left (x\right ) - \log \left (x^{2} + 1\right ) + 2 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 26, normalized size = 1.08 \begin {gather*} \frac {x \arctan \left (x\right ) - x \log \left (x^{2} + 1\right ) + 2 \, x \log \left (x - 1\right ) - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.79 \begin {gather*} 2 \log {\left (x - 1 \right )} - \log {\left (x^{2} + 1 \right )} + \operatorname {atan}{\left (x \right )} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.49, size = 23, normalized size = 0.96 \begin {gather*} -\frac {1}{x} + \arctan \left (x\right ) - \log \left (x^{2} + 1\right ) + 2 \, \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 = 2.13, size = 30, normalized size = 1.25 \begin {gather*} 2\,\ln \left (x-1\right )-\frac {1}{x}+\ln \left (x-\mathrm {i}\right )\,\left (-1-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-1+\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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