Optimal. Leaf size=48 \[ \frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{12 c^3}+\frac{b x^4}{12 c} \]
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Rubi [A] time = 0.0347672, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6097, 266, 43} \[ \frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{12 c^3}+\frac{b x^4}{12 c} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac{1}{3} (b c) \int \frac{x^7}{1-c^2 x^4} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^4\right )\\ &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^4\right )\\ &=\frac{b x^4}{12 c}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )+\frac{b \log \left (1-c^2 x^4\right )}{12 c^3}\\ \end{align*}
Mathematica [A] time = 0.0139929, size = 53, normalized size = 1.1 \[ \frac{a x^6}{6}+\frac{b \log \left (1-c^2 x^4\right )}{12 c^3}+\frac{b x^4}{12 c}+\frac{1}{6} b x^6 \tanh ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 45, normalized size = 0.9 \begin{align*}{\frac{{x}^{6}a}{6}}+{\frac{b{x}^{6}{\it Artanh} \left ( c{x}^{2} \right ) }{6}}+{\frac{b{x}^{4}}{12\,c}}+{\frac{b\ln \left ({c}^{2}{x}^{4}-1 \right ) }{12\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98235, size = 62, normalized size = 1.29 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{12} \,{\left (2 \, x^{6} \operatorname{artanh}\left (c x^{2}\right ) +{\left (\frac{x^{4}}{c^{2}} + \frac{\log \left (c^{2} x^{4} - 1\right )}{c^{4}}\right )} c\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05896, size = 134, normalized size = 2.79 \begin{align*} \frac{b c^{3} x^{6} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c^{3} x^{6} + b c^{2} x^{4} + b \log \left (c^{2} x^{4} - 1\right )}{12 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.0352, size = 85, normalized size = 1.77 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{atanh}{\left (c x^{2} \right )}}{6} + \frac{b x^{4}}{12 c} + \frac{b \log{\left (x - i \sqrt{\frac{1}{c}} \right )}}{6 c^{3}} + \frac{b \log{\left (x + i \sqrt{\frac{1}{c}} \right )}}{6 c^{3}} - \frac{b \operatorname{atanh}{\left (c x^{2} \right )}}{6 c^{3}} & \text{for}\: c \neq 0 \\\frac{a x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17006, size = 77, normalized size = 1.6 \begin{align*} \frac{1}{12} \, b x^{6} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + \frac{1}{6} \, a x^{6} + \frac{b x^{4}}{12 \, c} + \frac{b \log \left (c^{2} x^{4} - 1\right )}{12 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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