Optimal. Leaf size=117 \[ \frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{20 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right )}{10 c^{5/3}}+\frac{3 b x^2}{10 c} \]
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Rubi [A] time = 0.0972647, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6097, 275, 321, 200, 31, 634, 617, 204, 628} \[ \frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{20 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right )}{10 c^{5/3}}+\frac{3 b x^2}{10 c} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 275
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int x^4 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{5} (3 b c) \int \frac{x^7}{1-c^2 x^6} \, dx\\ &=\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{1}{10} (3 b c) \operatorname{Subst}\left (\int \frac{x^3}{1-c^2 x^3} \, dx,x,x^2\right )\\ &=\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^3} \, dx,x,x^2\right )}{10 c}\\ &=\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x} \, dx,x,x^2\right )}{10 c}-\frac{b \operatorname{Subst}\left (\int \frac{2+c^{2/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{10 c}\\ &=\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{c^{2/3}+2 c^{4/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{20 c^{5/3}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{20 c}\\ &=\frac{3 b x^2}{10 c}+\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 c^{2/3} x^2\right )}{10 c^{5/3}}\\ &=\frac{3 b x^2}{10 c}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1+2 c^{2/3} x^2}{\sqrt{3}}\right )}{10 c^{5/3}}+\frac{1}{5} x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^{2/3} x^2\right )}{10 c^{5/3}}-\frac{b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0282765, size = 198, normalized size = 1.69 \[ \frac{a x^5}{5}-\frac{b \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{20 c^{5/3}}-\frac{b \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{20 c^{5/3}}+\frac{b \log \left (1-\sqrt [3]{c} x\right )}{10 c^{5/3}}+\frac{b \log \left (\sqrt [3]{c} x+1\right )}{10 c^{5/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x-1}{\sqrt{3}}\right )}{10 c^{5/3}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right )}{10 c^{5/3}}+\frac{3 b x^2}{10 c}+\frac{1}{5} b x^5 \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 114, normalized size = 1. \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{{x}^{5}b{\it Artanh} \left ( c{x}^{3} \right ) }{5}}+{\frac{3\,b{x}^{2}}{10\,c}}+{\frac{b}{10\,{c}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{20\,{c}^{3}}\ln \left ({x}^{4}+\sqrt [3]{{c}^{-2}}{x}^{2}+ \left ({c}^{-2} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}}-{\frac{b\sqrt{3}}{10\,{c}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{{x}^{2}}{\sqrt [3]{{c}^{-2}}}}+1 \right ) } \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50079, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{5} \, a x^{5} + \frac{1}{20} \,{\left (4 \, x^{5} \operatorname{artanh}\left (c x^{3}\right ) + c{\left (\frac{6 \, x^{2}}{c^{2}} - \frac{2 \, \sqrt{3}{\left (c^{2}\right )}^{\frac{2}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}}{\left (2 \, x^{2} + \frac{1}{c^{2}}^{\frac{1}{3}}\right )}\right )}{c^{4}} - \frac{{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (x^{4} + \frac{1}{c^{2}}^{\frac{1}{3}} x^{2} + \frac{1}{c^{2}}^{\frac{2}{3}}\right )}{c^{4}} + \frac{2 \,{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (x^{2} - \frac{1}{c^{2}}^{\frac{1}{3}}\right )}{c^{4}}\right )}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79069, size = 414, normalized size = 3.54 \begin{align*} \frac{2 \, b c^{3} x^{5} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{3} x^{5} + 6 \, b c^{2} x^{2} - 2 \, \sqrt{3} b{\left (c^{2}\right )}^{\frac{1}{6}} c \arctan \left (-\frac{\sqrt{3}{\left (4 \, c^{2} x^{4} - 2 \,{\left (c^{2}\right )}^{\frac{2}{3}} x^{2} +{\left (c^{2}\right )}^{\frac{1}{3}}\right )}{\left (c^{2}\right )}^{\frac{1}{6}}}{8 \, c^{3} x^{6} + c}\right ) - b{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (c^{2} x^{4} +{\left (c^{2}\right )}^{\frac{2}{3}} x^{2} +{\left (c^{2}\right )}^{\frac{1}{3}}\right ) + 2 \, b{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (c^{2} x^{2} -{\left (c^{2}\right )}^{\frac{2}{3}}\right )}{20 \, 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 = 1.18607, size = 170, normalized size = 1.45 \begin{align*} -\frac{1}{20} \, b c^{9}{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{\left | c \right |}^{\frac{2}{3}}\right )}{c^{10}{\left | c \right |}^{\frac{2}{3}}} + \frac{\log \left (x^{4} + \frac{x^{2}}{{\left | c \right |}^{\frac{2}{3}}} + \frac{1}{{\left | c \right |}^{\frac{4}{3}}}\right )}{c^{10}{\left | c \right |}^{\frac{2}{3}}} - \frac{2 \, \log \left ({\left | x^{2} - \frac{1}{{\left | c \right |}^{\frac{2}{3}}} \right |}\right )}{c^{10}{\left | c \right |}^{\frac{2}{3}}}\right )} + \frac{1}{10} \, b x^{5} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{1}{5} \, a x^{5} + \frac{3 \, b x^{2}}{10 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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