Optimal. Leaf size=108 \[ -\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c}+x \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac{3 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.217234, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5910, 5984, 5918, 5948, 6058, 6610} \[ -\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c}+x \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac{3 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5910
Rule 5984
Rule 5918
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=x \left (a+b \tanh ^{-1}(c x)\right )^3-(3 b c) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^3-(3 b) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c}+\left (6 b^2\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c}-\frac{3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c}+\left (3 b^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{c}+x \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c}-\frac{3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c}+\frac{3 b^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.260217, size = 161, normalized size = 1.49 \[ \frac{6 a b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )+b^3 \left (6 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+2 \tanh ^{-1}(c x)^2 \left ((c x-1) \tanh ^{-1}(c x)-3 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )+3 a^2 b \log \left (1-c^2 x^2\right )+6 a^2 b c x \tanh ^{-1}(c x)+2 a^3 c x}{2 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.082, size = 261, normalized size = 2.4 \begin{align*} x{a}^{3}+{b}^{3}x \left ({\it Artanh} \left ( cx \right ) \right ) ^{3}+{\frac{{b}^{3} \left ({\it Artanh} \left ( cx \right ) \right ) ^{3}}{c}}-3\,{\frac{{b}^{3} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}}{c}\ln \left ({\frac{ \left ( cx+1 \right ) ^{2}}{-{c}^{2}{x}^{2}+1}}+1 \right ) }-3\,{\frac{{b}^{3}{\it Artanh} \left ( cx \right ) }{c}{\it polylog} \left ( 2,-{\frac{ \left ( cx+1 \right ) ^{2}}{-{c}^{2}{x}^{2}+1}} \right ) }+{\frac{3\,{b}^{3}}{2\,c}{\it polylog} \left ( 3,-{\frac{ \left ( cx+1 \right ) ^{2}}{-{c}^{2}{x}^{2}+1}} \right ) }+3\,xa{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}+3\,{\frac{a{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}}{c}}-6\,{\frac{{\it Artanh} \left ( cx \right ) a{b}^{2}}{c}\ln \left ({\frac{ \left ( cx+1 \right ) ^{2}}{-{c}^{2}{x}^{2}+1}}+1 \right ) }-3\,{\frac{a{b}^{2}}{c}{\it polylog} \left ( 2,-{\frac{ \left ( cx+1 \right ) ^{2}}{-{c}^{2}{x}^{2}+1}} \right ) }+3\,x{a}^{2}b{\it Artanh} \left ( cx \right ) +{\frac{3\,{a}^{2}b\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{2\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} x + \frac{3 \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a^{2} b}{2 \, c} - \frac{{\left (b^{3} c x - b^{3}\right )} \log \left (-c x + 1\right )^{3} - 3 \,{\left (2 \, a b^{2} c x +{\left (b^{3} c x + b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{8 \, c} - \int -\frac{{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{3} + 6 \,{\left (a b^{2} c x - a b^{2}\right )} \log \left (c x + 1\right )^{2} - 3 \,{\left (4 \, a b^{2} c x +{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{2} - 2 \,{\left (2 \, a b^{2} - b^{3} -{\left (2 \, a b^{2} c + b^{3} c\right )} x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \,{\left (c x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} \operatorname{artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{artanh}\left (c x\right ) + a^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]