Optimal. Leaf size=131 \[ \frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac{3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac{3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \]
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Rubi [A] time = 0.0874151, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2171, 2163, 2160, 2157, 29} \[ \frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac{3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac{3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2163
Rule 2160
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx &=\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac{(3 b) \int \frac{1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx}{-b x+\coth ^{-1}(\tanh (a+b x))}\\ &=-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac{(3 b) \int \frac{1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}+\frac{(3 b) \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 )^2}\\ &=-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac{(3 b) \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 )^2}+\frac{\left (3 b^2\right ) \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 )^2}\\ &=-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac{(3 b) \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 )^2}\\ &=-\frac{3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac{1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac{3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac{3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac{3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}\\ \end{align*}
Mathematica [A] time = 0.050491, size = 93, normalized size = 0.71 \[ -\frac{-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+2 \coth ^{-1}(\tanh (a+b x))^3+3 b x \coth ^{-1}(\tanh (a+b x))^2 \left (-2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )+2 \log (x)+1\right )+b^3 x^3}{2 x \coth ^{-1}(\tanh (a+b x))^2 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ({\rm arccoth} \left (\tanh \left ( bx+a \right ) \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.5656, size = 328, normalized size = 2.5 \begin{align*} \frac{48 \, b \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac{48 \, b \log \left (x\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac{8 \,{\left (12 \, b^{2} x^{2} - \pi ^{2} - 4 i \, \pi a + 4 \, a^{2} +{\left (-9 i \, \pi b + 18 \, a b\right )} x\right )}}{{\left (-4 i \, \pi ^{3} b^{2} + 24 \, \pi ^{2} a b^{2} + 48 i \, \pi a^{2} b^{2} - 32 \, a^{3} b^{2}\right )} x^{3} -{\left (4 \, \pi ^{4} b + 32 i \, \pi ^{3} a b - 96 \, \pi ^{2} a^{2} b - 128 i \, \pi a^{3} b + 64 \, a^{4} b\right )} x^{2} +{\left (i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11983, size = 2475, normalized size = 18.89 \begin{align*} \text{result too large to display} \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^{2} \operatorname{acoth}^{3}{\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^{2} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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