Optimal. Leaf size=48 \[ \frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac{b x^6}{18 c} \]
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Rubi [A] time = 0.0359128, 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} \[ \frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac{b x^6}{18 c} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{3} (b c) \int \frac{x^{11}}{1-c^2 x^6} \, dx\\ &=\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^6\right )\\ &=\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{18} (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^6\right )\\ &=\frac{b x^6}{18 c}+\frac{1}{9} x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}\\ \end{align*}
Mathematica [A] time = 0.0158537, size = 53, normalized size = 1.1 \[ \frac{a x^9}{9}+\frac{b \log \left (1-c^2 x^6\right )}{18 c^3}+\frac{b x^6}{18 c}+\frac{1}{9} b x^9 \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 45, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}a}{9}}+{\frac{b{x}^{9}{\it Artanh} \left ( c{x}^{3} \right ) }{9}}+{\frac{b{x}^{6}}{18\,c}}+{\frac{b\ln \left ({c}^{2}{x}^{6}-1 \right ) }{18\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02131, size = 62, normalized size = 1.29 \begin{align*} \frac{1}{9} \, a x^{9} + \frac{1}{18} \,{\left (2 \, x^{9} \operatorname{artanh}\left (c x^{3}\right ) +{\left (\frac{x^{6}}{c^{2}} + \frac{\log \left (c^{2} x^{6} - 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 = 1.91646, size = 134, normalized size = 2.79 \begin{align*} \frac{b c^{3} x^{9} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a c^{3} x^{9} + b c^{2} x^{6} + b \log \left (c^{2} x^{6} - 1\right )}{18 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19606, size = 77, normalized size = 1.6 \begin{align*} \frac{1}{18} \, b x^{9} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{1}{9} \, a x^{9} + \frac{b x^{6}}{18 \, c} + \frac{b \log \left (c^{2} x^{6} - 1\right )}{18 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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