Optimal. Leaf size=81 \[ -\frac{\text{PolyLog}\left (2,-\frac{a+b x+1}{-a-b x+1}\right )}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}+\frac{\tanh ^{-1}(a+b x)^2}{b}-\frac{2 \log \left (\frac{2}{-a-b x+1}\right ) \tanh ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.086324, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6103, 5910, 5984, 5918, 2402, 2315} \[ -\frac{\text{PolyLog}\left (2,-\frac{a+b x+1}{-a-b x+1}\right )}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}+\frac{\tanh ^{-1}(a+b x)^2}{b}-\frac{2 \log \left (\frac{2}{-a-b x+1}\right ) \tanh ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6103
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \tanh ^{-1}(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \tanh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac{2 \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\tanh ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac{2 \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{b}\\ &=\frac{\tanh ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac{2 \tanh ^{-1}(a+b x) \log \left (\frac{2}{1-a-b x}\right )}{b}+\frac{2 \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\tanh ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac{2 \tanh ^{-1}(a+b x) \log \left (\frac{2}{1-a-b x}\right )}{b}-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a-b x}\right )}{b}\\ &=\frac{\tanh ^{-1}(a+b x)^2}{b}+\frac{(a+b x) \tanh ^{-1}(a+b x)^2}{b}-\frac{2 \tanh ^{-1}(a+b x) \log \left (\frac{2}{1-a-b x}\right )}{b}-\frac{\text{Li}_2\left (1-\frac{2}{1-a-b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0216881, size = 55, normalized size = 0.68 \[ \frac{\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a+b x)}\right )+\tanh ^{-1}(a+b x) \left ((a+b x-1) \tanh ^{-1}(a+b x)-2 \log \left (e^{-2 \tanh ^{-1}(a+b x)}+1\right )\right )}{b} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.075, size = 103, normalized size = 1.3 \begin{align*} x \left ({\it Artanh} \left ( bx+a \right ) \right ) ^{2}+{\frac{ \left ({\it Artanh} \left ( bx+a \right ) \right ) ^{2}a}{b}}+{\frac{ \left ({\it Artanh} \left ( bx+a \right ) \right ) ^{2}}{b}}-2\,{\frac{{\it Artanh} \left ( bx+a \right ) }{b}\ln \left ({\frac{ \left ( bx+a+1 \right ) ^{2}}{1- \left ( bx+a \right ) ^{2}}}+1 \right ) }-{\frac{1}{b}{\it polylog} \left ( 2,-{\frac{ \left ( bx+a+1 \right ) ^{2}}{1- \left ( bx+a \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986349, size = 188, normalized size = 2.32 \begin{align*} -\frac{1}{4} \, b^{2}{\left (\frac{{\left (a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \,{\left (a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) +{\left (a - 1\right )} \log \left (b x + a - 1\right )^{2}}{b^{3}} + \frac{4 \,{\left (\log \left (b x + a - 1\right ) \log \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, b x - \frac{1}{2} \, a + \frac{1}{2}\right )\right )}}{b^{3}}\right )} + b{\left (\frac{{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac{{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} \operatorname{artanh}\left (b x + a\right ) + x \operatorname{artanh}\left (b x + a\right )^{2} \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 (\operatorname{artanh}\left (b x + a\right )^{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 \operatorname{atanh}^{2}{\left (a + b x \right )}\, 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 \operatorname{artanh}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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