Optimal. Leaf size=96 \[ -\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x^3}\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{3 c}-\frac{2 b \log \left (\frac{2}{1-c x^3}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{3 c} \]
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Rubi [B] time = 0.591888, antiderivative size = 207, normalized size of antiderivative = 2.16, number of steps used = 28, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6099, 2454, 2389, 2296, 2295, 6715, 2430, 43, 2416, 2394, 2393, 2391} \[ -\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )}{6 c}+\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )}{6 c}+\frac{1}{6} b x^3 \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c}+\frac{b \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{6 c}+\frac{b^2 \left (c x^3+1\right ) \log ^2\left (c x^3+1\right )}{12 c}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )}{6 c} \]
Warning: Unable to verify antiderivative.
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Rule 6099
Rule 2454
Rule 2389
Rule 2296
Rule 2295
Rule 6715
Rule 2430
Rule 43
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{2} b x^2 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{1}{4} b^2 x^2 \log ^2\left (1+c x^3\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \, dx-\frac{1}{2} b \int x^2 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right ) \, dx+\frac{1}{4} b^2 \int x^2 \log ^2\left (1+c x^3\right ) \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int (-2 a+b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )+\frac{1}{12} b^2 \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )\\ &=\frac{1}{6} b x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{12 c}+\frac{b^2 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{12 c}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{x (-2 a+b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac{1}{6} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x \log (1+c x)}{1-c x} \, dx,x,x^3\right )\\ &=-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c}+\frac{1}{6} b x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c}-\frac{b \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{6 c}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{6 c}+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{-2 a+b \log (1-c x)}{c}-\frac{-2 a+b \log (1-c x)}{c (1+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{6} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c}-\frac{\log (1+c x)}{c (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{3} a b x^3+\frac{b^2 x^3}{6}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c}-\frac{b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{6 c}+\frac{1}{6} b x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c}+\frac{1}{6} b \operatorname{Subst}\left (\int (-2 a+b \log (1-c x)) \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )+\frac{1}{6} b^2 \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )+\frac{1}{6} b^2 \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{6 c}\\ &=\frac{b^2 x^3}{3}+\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{6 c}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c}+\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{6 c}-\frac{b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{6 c}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{6 c}+\frac{1}{6} b x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c}-\frac{1}{6} b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )+\frac{1}{6} b^2 \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )-\frac{1}{6} b^2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{6 c}\\ &=\frac{b^2 x^3}{6}+\frac{b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{6 c}-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c}+\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{6 c}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{6 c}+\frac{1}{6} b x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{6 c}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{6 c}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{6 c}\\ &=-\frac{\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{12 c}+\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )}{6 c}+\frac{b^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{6 c}+\frac{1}{6} b x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac{b^2 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{12 c}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )}{6 c}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )}{6 c}\\ \end{align*}
Mathematica [A] time = 0.152375, size = 99, normalized size = 1.03 \[ \frac{b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+a \left (a c x^3+b \log \left (1-c^2 x^6\right )\right )+2 b \tanh ^{-1}\left (c x^3\right ) \left (a c x^3-b \log \left (e^{-2 \tanh ^{-1}\left (c x^3\right )}+1\right )\right )+b^2 \left (c x^3-1\right ) \tanh ^{-1}\left (c x^3\right )^2}{3 c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.004, size = 145, normalized size = 1.5 \begin{align*}{\frac{ \left ({\it Artanh} \left ( c{x}^{3} \right ) \right ) ^{2}{x}^{3}{b}^{2}}{3}}+{\frac{2\,ab{x}^{3}{\it Artanh} \left ( c{x}^{3} \right ) }{3}}+{\frac{{x}^{3}{a}^{2}}{3}}-{\frac{2\,{\it Artanh} \left ( c{x}^{3} \right ){b}^{2}}{3\,c}\ln \left ({\frac{ \left ( c{x}^{3}+1 \right ) ^{2}}{-{c}^{2}{x}^{6}+1}}+1 \right ) }+{\frac{{b}^{2} \left ({\it Artanh} \left ( c{x}^{3} \right ) \right ) ^{2}}{3\,c}}+{\frac{ab\ln \left ( -{c}^{2}{x}^{6}+1 \right ) }{3\,c}}-{\frac{{b}^{2}}{3\,c}{\it polylog} \left ( 2,-{\frac{ \left ( c{x}^{3}+1 \right ) ^{2}}{-{c}^{2}{x}^{6}+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} x^{3} + \frac{1}{12} \,{\left (x^{3} \log \left (-c x^{3} + 1\right )^{2} - c^{2}{\left (\frac{2 \, x^{3}}{c^{2}} - \frac{\log \left (c x^{3} + 1\right )}{c^{3}} + \frac{\log \left (c x^{3} - 1\right )}{c^{3}}\right )} - 2 \,{\left (\frac{x^{3}}{c} + \frac{\log \left (c x^{3} - 1\right )}{c^{2}}\right )} c \log \left (-c x^{3} + 1\right ) + 18 \, c \int \frac{x^{5} \log \left (c x^{3} + 1\right )}{c^{2} x^{6} - 1}\,{d x} + \frac{c x^{3} \log \left (c x^{3} + 1\right )^{2} + 2 \,{\left (c x^{3} -{\left (c x^{3} + 1\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )}{c} + \frac{2 \, c x^{3} + \log \left (c x^{3} - 1\right )^{2} + 2 \, \log \left (c x^{3} - 1\right )}{c} - \frac{\log \left (c^{2} x^{6} - 1\right )}{c} + 6 \, \int \frac{x^{2} \log \left (c x^{3} + 1\right )}{c^{2} x^{6} - 1}\,{d x}\right )} b^{2} + \frac{{\left (2 \, c x^{3} \operatorname{artanh}\left (c x^{3}\right ) + \log \left (-c^{2} x^{6} + 1\right )\right )} a b}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{2} \operatorname{artanh}\left (c x^{3}\right )^{2} + 2 \, a b x^{2} \operatorname{artanh}\left (c x^{3}\right ) + a^{2} x^{2}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{3}\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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