Optimal. Leaf size=24 \[ \frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0180047, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1876, 212, 206, 203, 275} \[ \frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 1876
Rule 212
Rule 206
Rule 203
Rule 275
Rubi steps
\begin{align*} \int \frac{c+d x}{1-x^4} \, dx &=\int \left (\frac{c}{1-x^4}+\frac{d x}{1-x^4}\right ) \, dx\\ &=c \int \frac{1}{1-x^4} \, dx+d \int \frac{x}{1-x^4} \, dx\\ &=\frac{1}{2} c \int \frac{1}{1-x^2} \, dx+\frac{1}{2} c \int \frac{1}{1+x^2} \, dx+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0147418, size = 42, normalized size = 1.75 \[ \frac{1}{4} \left (-(c+d) \log (1-x)+c \log (x+1)+2 c \tan ^{-1}(x)+d \log \left (x^2+1\right )-d \log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 44, normalized size = 1.8 \begin{align*} -{\frac{\ln \left ( -1+x \right ) c}{4}}-{\frac{\ln \left ( -1+x \right ) d}{4}}+{\frac{\ln \left ( 1+x \right ) c}{4}}-{\frac{\ln \left ( 1+x \right ) d}{4}}+{\frac{d\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{c\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4422, size = 47, normalized size = 1.96 \begin{align*} \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left (x + 1\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29288, size = 119, normalized size = 4.96 \begin{align*} \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left (x + 1\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.459214, size = 313, normalized size = 13.04 \begin{align*} \frac{\left (c - d\right ) \log{\left (x + \frac{c^{4} \left (c - d\right ) + 5 c^{2} d^{3} + c^{2} d \left (c - d\right )^{2} - 2 d^{4} \left (c - d\right ) + 2 d^{2} \left (c - d\right )^{3}}{c^{5} + 4 c d^{4}} \right )}}{4} - \frac{\left (c + d\right ) \log{\left (x + \frac{- c^{4} \left (c + d\right ) + 5 c^{2} d^{3} + c^{2} d \left (c + d\right )^{2} + 2 d^{4} \left (c + d\right ) - 2 d^{2} \left (c + d\right )^{3}}{c^{5} + 4 c d^{4}} \right )}}{4} - \left (- \frac{i c}{4} - \frac{d}{4}\right ) \log{\left (x + \frac{- 4 c^{4} \left (- \frac{i c}{4} - \frac{d}{4}\right ) + 5 c^{2} d^{3} + 16 c^{2} d \left (- \frac{i c}{4} - \frac{d}{4}\right )^{2} + 8 d^{4} \left (- \frac{i c}{4} - \frac{d}{4}\right ) - 128 d^{2} \left (- \frac{i c}{4} - \frac{d}{4}\right )^{3}}{c^{5} + 4 c d^{4}} \right )} - \left (\frac{i c}{4} - \frac{d}{4}\right ) \log{\left (x + \frac{- 4 c^{4} \left (\frac{i c}{4} - \frac{d}{4}\right ) + 5 c^{2} d^{3} + 16 c^{2} d \left (\frac{i c}{4} - \frac{d}{4}\right )^{2} + 8 d^{4} \left (\frac{i c}{4} - \frac{d}{4}\right ) - 128 d^{2} \left (\frac{i c}{4} - \frac{d}{4}\right )^{3}}{c^{5} + 4 c d^{4}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05346, size = 50, normalized size = 2.08 \begin{align*} \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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