Optimal. Leaf size=50 \[ \frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{4} b c^3 x-\frac{1}{4} b c^4 \tanh ^{-1}\left (\frac{x}{c}\right )+\frac{1}{12} b c x^3 \]
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Rubi [A] time = 0.0319407, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6097, 263, 302, 207} \[ \frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{4} b c^3 x-\frac{1}{4} b c^4 \tanh ^{-1}\left (\frac{x}{c}\right )+\frac{1}{12} b c x^3 \]
Antiderivative was successfully verified.
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Rule 6097
Rule 263
Rule 302
Rule 207
Rubi steps
\begin{align*} \int x^3 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{4} (b c) \int \frac{x^2}{1-\frac{c^2}{x^2}} \, dx\\ &=\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{4} (b c) \int \frac{x^4}{-c^2+x^2} \, dx\\ &=\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{4} (b c) \int \left (c^2+x^2+\frac{c^4}{-c^2+x^2}\right ) \, dx\\ &=\frac{1}{4} b c^3 x+\frac{1}{12} b c x^3+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )+\frac{1}{4} \left (b c^5\right ) \int \frac{1}{-c^2+x^2} \, dx\\ &=\frac{1}{4} b c^3 x+\frac{1}{12} b c x^3+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )-\frac{1}{4} b c^4 \tanh ^{-1}\left (\frac{x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0099763, size = 67, normalized size = 1.34 \[ \frac{a x^4}{4}+\frac{1}{4} b c^3 x+\frac{1}{8} b c^4 \log (x-c)-\frac{1}{8} b c^4 \log (c+x)+\frac{1}{12} b c x^3+\frac{1}{4} b x^4 \tanh ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 62, normalized size = 1.2 \begin{align*}{\frac{{x}^{4}a}{4}}+{\frac{b{x}^{4}}{4}{\it Artanh} \left ({\frac{c}{x}} \right ) }+{\frac{{c}^{4}b}{8}\ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{bc{x}^{3}}{12}}+{\frac{b{c}^{3}x}{4}}-{\frac{{c}^{4}b}{8}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978632, size = 77, normalized size = 1.54 \begin{align*} \frac{1}{4} \, a x^{4} + \frac{1}{24} \,{\left (6 \, x^{4} \operatorname{artanh}\left (\frac{c}{x}\right ) -{\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66774, size = 113, normalized size = 2.26 \begin{align*} \frac{1}{4} \, b c^{3} x + \frac{1}{12} \, b c x^{3} + \frac{1}{4} \, a x^{4} - \frac{1}{8} \,{\left (b c^{4} - b x^{4}\right )} \log \left (-\frac{c + x}{c - x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.75908, size = 46, normalized size = 0.92 \begin{align*} \frac{a x^{4}}{4} - \frac{b c^{4} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{4} + \frac{b c^{3} x}{4} + \frac{b c x^{3}}{12} + \frac{b x^{4} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16809, size = 84, normalized size = 1.68 \begin{align*} -\frac{1}{8} \, b c^{4} \log \left (c + x\right ) + \frac{1}{8} \, b c^{4} \log \left (c - x\right ) + \frac{1}{8} \, b x^{4} \log \left (-\frac{c + x}{c - x}\right ) + \frac{1}{4} \, b c^{3} x + \frac{1}{12} \, b c x^{3} + \frac{1}{4} \, a x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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