Optimal. Leaf size=92 \[ -\frac{b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac{b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac{b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0628528, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2163, 2160, 2157, 29} \[ -\frac{b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac{b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac{b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2163
Rule 2160
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x^3 \coth ^{-1}(\tanh (a+b x))} \, dx &=\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac{b \int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))} \, dx}{-b x+\coth ^{-1}(\tanh (a+b x))}\\ &=\frac{b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac{b^2 \int \frac{1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}+\frac{b^2 \int \frac{1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}-\frac{b^3 \int \frac{1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac{b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac{1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac{b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac{b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0270553, size = 66, normalized size = 0.72 \[ \frac{b^2 x^2 \left (2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )-2 \log (x)+3\right )-4 b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}{\rm arccoth} \left (\tanh \left ( bx+a \right ) \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.79593, size = 146, normalized size = 1.59 \begin{align*} \frac{8 \, b^{2} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} - \frac{8 \, b^{2} \log \left (x\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} - \frac{2 \,{\left (i \, \pi + 4 \, b x - 2 \, a\right )}}{{\left (2 \, \pi ^{2} + 8 i \, \pi a - 8 \, a^{2}\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71336, size = 448, normalized size = 4.87 \begin{align*} -\frac{2 \,{\left (\pi ^{4} a + 8 \, \pi ^{2} a^{3} + 16 \, a^{5} - 8 \,{\left (\pi ^{3} b^{2} - 12 \, \pi a^{2} b^{2}\right )} x^{2} \arctan \left (-\frac{2 \, b x + 2 \, a - \sqrt{4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) - 4 \,{\left (3 \, \pi ^{2} a b^{2} - 4 \, a^{3} b^{2}\right )} x^{2} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) + 8 \,{\left (3 \, \pi ^{2} a b^{2} - 4 \, a^{3} b^{2}\right )} x^{2} \log \left (x\right ) + 2 \,{\left (\pi ^{4} b - 16 \, a^{4} b\right )} x\right )}}{{\left (\pi ^{6} + 12 \, \pi ^{4} a^{2} + 48 \, \pi ^{2} a^{4} + 64 \, a^{6}\right )} x^{2}} \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{1}{x^{3} \operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \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 \frac{1}{x^{3} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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