Optimal. Leaf size=45 \[ \frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.10, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1593, 6725, 212, 206, 203, 15, 298} \begin {gather*} \frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 203
Rule 206
Rule 212
Rule 298
Rule 1593
Rule 6725
Rubi steps
\begin {align*} \int \frac {x-\sqrt {x^6}}{x-x^5} \, dx &=\int \frac {x-\sqrt {x^6}}{x \left (1-x^4\right )} \, dx\\ &=\int \left (\frac {1}{1-x^4}+\frac {\sqrt {x^6}}{x \left (-1+x^4\right )}\right ) \, dx\\ &=\int \frac {1}{1-x^4} \, dx+\int \frac {\sqrt {x^6}}{x \left (-1+x^4\right )} \, dx\\ &=\frac {1}{2} \int \frac {1}{1-x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\frac {\sqrt {x^6} \int \frac {x^2}{-1+x^4} \, dx}{x^3}\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {x^6} \int \frac {1}{1-x^2} \, dx}{2 x^3}+\frac {\sqrt {x^6} \int \frac {1}{1+x^2} \, dx}{2 x^3}\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.60 \begin {gather*} \frac {1}{2} \left (\frac {\sqrt {x^6} \left (\tan ^{-1}(x)-\tanh ^{-1}(x)\right )}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.27, size = 57, normalized size = 1.27 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {x^6}}{x^4}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x^6}}{x^4}\right )-\frac {1}{4} \log (1-x)+\frac {1}{4} \log (x+1)+\frac {1}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 2, normalized size = 0.04 \begin {gather*} \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 31, normalized size = 0.69 \begin {gather*} \frac {1}{2} \, {\left (\mathrm {sgn}\relax (x) + 1\right )} \arctan \relax (x) - \frac {1}{4} \, {\left (\mathrm {sgn}\relax (x) - 1\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, {\left (\mathrm {sgn}\relax (x) - 1\right )} \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.78 \begin {gather*} \frac {\arctanh \relax (x )}{2}+\frac {\arctan \relax (x )}{2}+\frac {\sqrt {x^{6}}\, \left (2 \arctan \relax (x )+\ln \left (x -1\right )-\ln \left (x +1\right )\right )}{4 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 2, normalized size = 0.04 \begin {gather*} \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x-\sqrt {x^6}}{x-x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 2, normalized size = 0.04 \begin {gather*} \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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