Optimal. Leaf size=34 \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} b c \log \left (c^2-x^4\right ) \]
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Rubi [A] time = 0.0180373, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6097, 263, 260} \[ \frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} b c \log \left (c^2-x^4\right ) \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 260
Rubi steps
\begin{align*} \int x \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+(b c) \int \frac{1}{\left (1-\frac{c^2}{x^4}\right ) x} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+(b c) \int \frac{x^3}{-c^2+x^4} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )+\frac{1}{4} b c \log \left (c^2-x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0068589, size = 39, normalized size = 1.15 \[ \frac{a x^2}{2}+\frac{1}{4} b c \log \left (x^4-c^2\right )+\frac{1}{2} b x^2 \tanh ^{-1}\left (\frac{c}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 52, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}}{2}{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) }+{\frac{bc}{4}\ln \left ( 1+{\frac{c}{{x}^{2}}} \right ) }-bc\ln \left ({x}^{-1} \right ) +{\frac{bc}{4}\ln \left ({\frac{c}{{x}^{2}}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981602, size = 46, normalized size = 1.35 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + c \log \left (x^{4} - c^{2}\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64731, size = 99, normalized size = 2.91 \begin{align*} \frac{1}{4} \, b x^{2} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) + \frac{1}{2} \, a x^{2} + \frac{1}{4} \, b c \log \left (x^{4} - c^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.1228, size = 61, normalized size = 1.79 \begin{align*} \frac{a x^{2}}{2} + \frac{b c \log{\left (- i \sqrt{c} + x \right )}}{2} + \frac{b c \log{\left (i \sqrt{c} + x \right )}}{2} - \frac{b c \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} + \frac{b x^{2} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38631, size = 63, normalized size = 1.85 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{4} \,{\left (x^{2} \log \left (-\frac{\frac{c}{x^{2}} + 1}{\frac{c}{x^{2}} - 1}\right ) + c \log \left ({\left | x^{4} - c^{2} \right |}\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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