Optimal. Leaf size=197 \[ -\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c^3}+\frac{a b^2 x}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac{b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{b^3 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2} \]
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Rubi [A] time = 0.449067, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5916, 5980, 5910, 260, 5948, 5984, 5918, 6058, 6610} \[ -\frac{b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{2 c^3}+\frac{a b^2 x}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac{b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{b^3 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 5916
Rule 5980
Rule 5910
Rule 260
Rule 5948
Rule 5984
Rule 5918
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-(b c) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac{b \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}-\frac{b \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c}\\ &=\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-b^2 \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{b \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx}{c^2}\\ &=\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}+\frac{b^2 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac{b^2 \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}+\frac{\left (2 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}{c^2}\\ &=\frac{a b^2 x}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \int \tanh ^{-1}(c x) \, dx}{c^2}+\frac{b^3 \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac{a b^2 x}{c^2}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )}{2 c^3}-\frac{b^3 \int \frac{x}{1-c^2 x^2} \, dx}{c}\\ &=\frac{a b^2 x}{c^2}+\frac{b^3 x \tanh ^{-1}(c x)}{c^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac{b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac{b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac{b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^3}+\frac{b^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.520302, size = 250, normalized size = 1.27 \[ \frac{6 a b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\left (c^3 x^3-1\right ) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (c^2 x^2-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-1\right )+c x\right )+b^3 \left (6 \tanh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+2 c^3 x^3 \tanh ^{-1}(c x)^3+3 c^2 x^2 \tanh ^{-1}(c x)^2-2 \tanh ^{-1}(c x)^3-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)-6 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+3 a^2 b c^2 x^2+3 a^2 b \log \left (1-c^2 x^2\right )+6 a^2 b c^3 x^3 \tanh ^{-1}(c x)+2 a^3 c^3 x^3}{6 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.477, size = 1177, normalized size = 6. \begin{align*} \text{result too large to display} \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^{3} x^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a^{2} b - \frac{{\left (b^{3} c^{3} x^{3} - b^{3}\right )} \log \left (-c x + 1\right )^{3} - 3 \,{\left (2 \, a b^{2} c^{3} x^{3} + b^{3} c^{2} x^{2} +{\left (b^{3} c^{3} x^{3} + b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{24 \, c^{3}} - \int -\frac{{\left (b^{3} c^{3} x^{3} - b^{3} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{3} + 6 \,{\left (a b^{2} c^{3} x^{3} - a b^{2} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{2} -{\left (4 \, a b^{2} c^{3} x^{3} + 2 \, b^{3} c^{2} x^{2} + 3 \,{\left (b^{3} c^{3} x^{3} - b^{3} c^{2} x^{2}\right )} \log \left (c x + 1\right )^{2} - 2 \,{\left (6 \, a b^{2} c^{2} x^{2} -{\left (6 \, a b^{2} c^{3} + b^{3} c^{3}\right )} x^{3} - b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \,{\left (c^{3} x - c^{2}\right )}}\,{d x} \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^{3} x^{2} \operatorname{artanh}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname{artanh}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname{artanh}\left (c x\right ) + a^{3} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}\, dx \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\right ) + a\right )}^{3} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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