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 = 0.158017, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6725, 212, 206, 203, 15, 298} \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6725
Rule 212
Rule 206
Rule 203
Rule 15
Rule 298
Rubi steps
\begin{align*} \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*}
Mathematica [A] time = 0.0671234, size = 27, normalized size = 0.6 \[ \frac{1}{2} \left (\frac{\sqrt{x^6} \left (\tan ^{-1}(x)-\tanh ^{-1}(x)\right )}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 35, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x-1 \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{{x}^{6}}}+{\frac{{\it Artanh} \left ( x \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6714, size = 3, normalized size = 0.07 \begin{align*} \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65722, size = 15, normalized size = 0.33 \begin{align*} \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.096463, size = 2, normalized size = 0.04 \begin{align*} \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09433, size = 42, normalized size = 0.93 \begin{align*} \frac{1}{2} \,{\left (\mathrm{sgn}\left (x\right ) + 1\right )} \arctan \left (x\right ) - \frac{1}{4} \,{\left (\mathrm{sgn}\left (x\right ) - 1\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \,{\left (\mathrm{sgn}\left (x\right ) - 1\right )} \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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