Optimal. Leaf size=111 \[ \frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{c x+1}\right )}{4 c}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{c} \]
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Rubi [A] time = 0.221338, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {5918, 5948, 6056, 6060, 6610} \[ \frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{c x+1}\right )}{4 c}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{c} \]
Antiderivative was successfully verified.
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Rule 5918
Rule 5948
Rule 6056
Rule 6060
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{1+c x} \, dx &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+c x}\right )}{c}+(3 b) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+c x}\right )}{c}+\frac{3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c}-\left (3 b^2\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \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 \log \left (\frac{2}{1+c x}\right )}{c}+\frac{3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c}+\frac{3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c}-\frac{1}{2} \left (3 b^3\right ) \int \frac{\text{Li}_3\left (1-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3 \log \left (\frac{2}{1+c x}\right )}{c}+\frac{3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c}+\frac{3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{2 c}+\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1+c x}\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.287449, size = 152, normalized size = 1.37 \[ \frac{6 b^2 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right ) \left (a+b \tanh ^{-1}(c x)\right )+6 b \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+3 b^3 \text{PolyLog}\left (4,-e^{-2 \tanh ^{-1}(c x)}\right )-12 a^2 b \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+4 a^3 \log (c x+1)-12 a b^2 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-4 b^3 \tanh ^{-1}(c x)^3 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )}{4 c} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.313, size = 1491, normalized size = 13.4 \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{b^{3} \log \left (c x + 1\right ) \log \left (-c x + 1\right )^{3}}{8 \, c} + \frac{a^{3} \log \left (c x + 1\right )}{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} + 6 \,{\left (b^{3} c x \log \left (c x + 1\right ) + a b^{2} c x - a b^{2}\right )} \log \left (-c x + 1\right )^{2} + 12 \,{\left (a^{2} b c x - a^{2} b\right )} \log \left (c x + 1\right ) - 3 \,{\left (4 \, a^{2} b c x - 4 \, a^{2} b +{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b^{2} c x - a b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \,{\left (c^{2} x^{2} - 1\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 (\frac{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}}{c x + 1}, 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 \frac{\left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{3}}{c x + 1}\, 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 \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{3}}{c x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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