Optimal. Leaf size=54 \[ \frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b x^3}{12 c^3}-\frac{b \tanh ^{-1}\left (c x^3\right )}{12 c^4}+\frac{b x^9}{36 c} \]
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Rubi [A] time = 0.0382175, antiderivative size = 54, 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, 275, 302, 206} \[ \frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b x^3}{12 c^3}-\frac{b \tanh ^{-1}\left (c x^3\right )}{12 c^4}+\frac{b x^9}{36 c} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 275
Rule 302
Rule 206
Rubi steps
\begin{align*} \int x^{11} \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{4} (b c) \int \frac{x^{14}}{1-c^2 x^6} \, dx\\ &=\frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{x^4}{1-c^2 x^2} \, dx,x,x^3\right )\\ &=\frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx,x,x^3\right )\\ &=\frac{b x^3}{12 c^3}+\frac{b x^9}{36 c}+\frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,x^3\right )}{12 c^3}\\ &=\frac{b x^3}{12 c^3}+\frac{b x^9}{36 c}-\frac{b \tanh ^{-1}\left (c x^3\right )}{12 c^4}+\frac{1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0200856, size = 78, normalized size = 1.44 \[ \frac{a x^{12}}{12}+\frac{b x^3}{12 c^3}+\frac{b \log \left (1-c x^3\right )}{24 c^4}-\frac{b \log \left (c x^3+1\right )}{24 c^4}+\frac{b x^9}{36 c}+\frac{1}{12} b x^{12} \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 66, normalized size = 1.2 \begin{align*}{\frac{{x}^{12}a}{12}}+{\frac{b{x}^{12}{\it Artanh} \left ( c{x}^{3} \right ) }{12}}+{\frac{b{x}^{9}}{36\,c}}+{\frac{b{x}^{3}}{12\,{c}^{3}}}+{\frac{b\ln \left ( c{x}^{3}-1 \right ) }{24\,{c}^{4}}}-{\frac{b\ln \left ( c{x}^{3}+1 \right ) }{24\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953359, size = 93, normalized size = 1.72 \begin{align*} \frac{1}{12} \, a x^{12} + \frac{1}{72} \,{\left (6 \, x^{12} \operatorname{artanh}\left (c x^{3}\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{9} + 3 \, x^{3}\right )}}{c^{4}} - \frac{3 \, \log \left (c x^{3} + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x^{3} - 1\right )}{c^{5}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04169, size = 138, normalized size = 2.56 \begin{align*} \frac{6 \, a c^{4} x^{12} + 2 \, b c^{3} x^{9} + 6 \, b c x^{3} + 3 \,{\left (b c^{4} x^{12} - b\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{72 \, c^{4}} \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 = 1.20153, size = 105, normalized size = 1.94 \begin{align*} \frac{1}{24} \, b x^{12} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{1}{12} \, a x^{12} + \frac{b x^{9}}{36 \, c} + \frac{b x^{3}}{12 \, c^{3}} - \frac{b \log \left (c x^{3} + 1\right )}{24 \, c^{4}} + \frac{b \log \left (c x^{3} - 1\right )}{24 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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