Optimal. Leaf size=43 \[ \frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )-\frac{1}{4} b c^2 \tanh ^{-1}\left (\frac{x^2}{c}\right )+\frac{1}{4} b c x^2 \]
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Rubi [A] time = 0.0297014, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6097, 263, 275, 321, 207} \[ \frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )-\frac{1}{4} b c^2 \tanh ^{-1}\left (\frac{x^2}{c}\right )+\frac{1}{4} b c x^2 \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 275
Rule 321
Rule 207
Rubi steps
\begin{align*} \int x^3 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right ) \, dx &=\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{2} (b c) \int \frac{x}{1-\frac{c^2}{x^4}} \, dx\\ &=\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{2} (b c) \int \frac{x^5}{-c^2+x^4} \, dx\\ &=\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{x^2}{-c^2+x^2} \, dx,x,x^2\right )\\ &=\frac{1}{4} b c x^2+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2+x^2} \, dx,x,x^2\right )\\ &=\frac{1}{4} b c x^2+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )-\frac{1}{4} b c^2 \tanh ^{-1}\left (\frac{x^2}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0111031, size = 62, normalized size = 1.44 \[ \frac{a x^4}{4}+\frac{1}{8} b c^2 \log \left (x^2-c\right )-\frac{1}{8} b c^2 \log \left (c+x^2\right )+\frac{1}{4} b c x^2+\frac{1}{4} b x^4 \tanh ^{-1}\left (\frac{c}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 55, normalized size = 1.3 \begin{align*}{\frac{{x}^{4}a}{4}}+{\frac{b{x}^{4}}{4}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }-{\frac{b{c}^{2}}{8}\ln \left ( 1+{\frac{c}{{x}^{2}}} \right ) }+{\frac{b{c}^{2}}{8}\ln \left ({\frac{c}{{x}^{2}}}-1 \right ) }+{\frac{bc{x}^{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960415, size = 66, normalized size = 1.53 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{8} \,{\left (2 \, x^{4} \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) +{\left (2 \, x^{2} - c \log \left (x^{2} + c\right ) + c \log \left (x^{2} - c\right )\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.585, size = 97, normalized size = 2.26 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{4} \, b c x^{2} + \frac{1}{8} \,{\left (b x^{4} - b c^{2}\right )} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.3665, size = 41, normalized size = 0.95 \begin{align*} \frac{a x^{4}}{4} - \frac{b c^{2} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{4} + \frac{b c x^{2}}{4} + \frac{b x^{4} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25419, size = 84, normalized size = 1.95 \begin{align*} \frac{1}{8} \, b x^{4} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + \frac{1}{4} \, a x^{4} + \frac{1}{4} \, b c x^{2} - \frac{1}{8} \, b c^{2} \log \left (x^{2} + c\right ) + \frac{1}{8} \, b c^{2} \log \left (-x^{2} + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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