Optimal. Leaf size=117 \[ -b \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \]
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Rubi [A] time = 0.263671, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5914, 6052, 5948, 6058, 6610} \[ -b \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-(4 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+(2 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-(2 b c) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\left (b^2 c\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\frac{1}{2} b^2 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{1}{2} b^2 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [A] time = 0.0715302, size = 120, normalized size = 1.03 \[ \frac{1}{2} b \left (2 \text{PolyLog}\left (2,\frac{c x+1}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-2 \text{PolyLog}\left (2,\frac{c x+1}{c x-1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b \left (\text{PolyLog}\left (3,\frac{c x+1}{c x-1}\right )-\text{PolyLog}\left (3,\frac{c x+1}{1-c x}\right )\right )\right )+2 \tanh ^{-1}\left (\frac{c x+1}{c x-1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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Maple [C] time = 0.214, size = 701, 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*} a^{2} \log \left (x\right ) + \int \frac{b^{2}{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}^{2}}{4 \, x} + \frac{a b{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{x}\,{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^{2} \operatorname{artanh}\left (c x\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x\right ) + a^{2}}{x}, 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 )^{2}}{x}\, 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 )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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