Optimal. Leaf size=47 \[ \frac{1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (c^2 x^6+1\right )}{18 c^3}-\frac{b x^6}{18 c} \]
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Rubi [A] time = 0.036474, antiderivative size = 47, 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 = {5033, 266, 43} \[ \frac{1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (c^2 x^6+1\right )}{18 c^3}-\frac{b x^6}{18 c} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{9} x^9 \left (a+b \tan ^{-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 \tan ^{-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 \tan ^{-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 \tan ^{-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 = 0.0134892, size = 52, normalized size = 1.11 \[ \frac{a x^9}{9}+\frac{b \log \left (c^2 x^6+1\right )}{18 c^3}-\frac{b x^6}{18 c}+\frac{1}{9} b x^9 \tan ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 45, normalized size = 1. \begin{align*}{\frac{{x}^{9}a}{9}}+{\frac{b{x}^{9}\arctan \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 = 1.05226, size = 65, normalized size = 1.38 \begin{align*} \frac{1}{9} \, a x^{9} + \frac{1}{18} \,{\left (2 \, x^{9} \arctan \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 = 2.65607, size = 115, normalized size = 2.45 \begin{align*} \frac{2 \, b c^{3} x^{9} \arctan \left (c x^{3}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13653, size = 63, normalized size = 1.34 \begin{align*} \frac{2 \, a c x^{9} +{\left (2 \, c x^{9} \arctan \left (c x^{3}\right ) - x^{6} + \frac{\log \left (c^{2} x^{6} + 1\right )}{c^{2}}\right )} b}{18 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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