Optimal. Leaf size=97 \[ \frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2160
Rule 2163
Rubi steps
\begin {align*} \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {\int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))}\\ &=-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\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 {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {\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 \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 {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac {\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 {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 74, normalized size = 0.76 \[ \frac {-4 b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2 \left (-2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )+2 \log (b x)+3\right )+b^2 x^2}{2 \coth ^{-1}(\tanh (a+b x))^2 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 811, normalized size = 8.36 \[ -\frac {8 \, {\left (9 \, \pi ^{6} a + 60 \, \pi ^{4} a^{3} + 48 \, \pi ^{2} a^{5} - 192 \, a^{7} + 8 \, {\left (\pi ^{4} b^{3} - 16 \, a^{4} b^{3}\right )} x^{3} + 4 \, {\left (9 \, \pi ^{4} a b^{2} + 8 \, \pi ^{2} a^{3} b^{2} - 112 \, a^{5} b^{2}\right )} x^{2} + 4 \, {\left (\pi ^{6} b + 16 \, \pi ^{4} a^{2} b + 16 \, \pi ^{2} a^{4} b - 128 \, a^{6} b\right )} x - 2 \, {\left (\pi ^{7} - 4 \, \pi ^{5} a^{2} - 80 \, \pi ^{3} a^{4} - 192 \, \pi a^{6} + 16 \, {\left (\pi ^{3} b^{4} - 12 \, \pi a^{2} b^{4}\right )} x^{4} + 64 \, {\left (\pi ^{3} a b^{3} - 12 \, \pi a^{3} b^{3}\right )} x^{3} + 8 \, {\left (\pi ^{5} b^{2} - 144 \, \pi a^{4} b^{2}\right )} x^{2} + 16 \, {\left (\pi ^{5} a b - 8 \, \pi ^{3} a^{3} b - 48 \, \pi a^{5} b\right )} x\right )} \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 ) - {\left (3 \, \pi ^{6} a + 20 \, \pi ^{4} a^{3} + 16 \, \pi ^{2} a^{5} - 64 \, a^{7} + 16 \, {\left (3 \, \pi ^{2} a b^{4} - 4 \, a^{3} b^{4}\right )} x^{4} + 64 \, {\left (3 \, \pi ^{2} a^{2} b^{3} - 4 \, a^{4} b^{3}\right )} x^{3} + 8 \, {\left (3 \, \pi ^{4} a b^{2} + 32 \, \pi ^{2} a^{3} b^{2} - 48 \, a^{5} b^{2}\right )} x^{2} + 16 \, {\left (3 \, \pi ^{4} a^{2} b + 8 \, \pi ^{2} a^{4} b - 16 \, a^{6} b\right )} x\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) + 2 \, {\left (3 \, \pi ^{6} a + 20 \, \pi ^{4} a^{3} + 16 \, \pi ^{2} a^{5} - 64 \, a^{7} + 16 \, {\left (3 \, \pi ^{2} a b^{4} - 4 \, a^{3} b^{4}\right )} x^{4} + 64 \, {\left (3 \, \pi ^{2} a^{2} b^{3} - 4 \, a^{4} b^{3}\right )} x^{3} + 8 \, {\left (3 \, \pi ^{4} a b^{2} + 32 \, \pi ^{2} a^{3} b^{2} - 48 \, a^{5} b^{2}\right )} x^{2} + 16 \, {\left (3 \, \pi ^{4} a^{2} b + 8 \, \pi ^{2} a^{4} b - 16 \, a^{6} b\right )} x\right )} \log \relax (x)\right )}}{\pi ^{10} + 20 \, \pi ^{8} a^{2} + 160 \, \pi ^{6} a^{4} + 640 \, \pi ^{4} a^{6} + 1280 \, \pi ^{2} a^{8} + 1024 \, a^{10} + 16 \, {\left (\pi ^{6} b^{4} + 12 \, \pi ^{4} a^{2} b^{4} + 48 \, \pi ^{2} a^{4} b^{4} + 64 \, a^{6} b^{4}\right )} x^{4} + 64 \, {\left (\pi ^{6} a b^{3} + 12 \, \pi ^{4} a^{3} b^{3} + 48 \, \pi ^{2} a^{5} b^{3} + 64 \, a^{7} b^{3}\right )} x^{3} + 8 \, {\left (\pi ^{8} b^{2} + 24 \, \pi ^{6} a^{2} b^{2} + 192 \, \pi ^{4} a^{4} b^{2} + 640 \, \pi ^{2} a^{6} b^{2} + 768 \, a^{8} b^{2}\right )} x^{2} + 16 \, {\left (\pi ^{8} a b + 16 \, \pi ^{6} a^{3} b + 96 \, \pi ^{4} a^{5} b + 256 \, \pi ^{2} a^{7} b + 256 \, a^{9} b\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.13, size = 173, normalized size = 1.78 \[ \frac {8 \, {\left (-3 i \, \pi + 4 \, b x + 6 \, a\right )}}{2 \, \pi ^{4} + 16 i \, \pi ^{3} a - 48 \, \pi ^{2} a^{2} - 64 i \, \pi a^{3} + 32 \, a^{4} - {\left (8 \, \pi ^{2} b^{2} + 32 i \, \pi a b^{2} - 32 \, a^{2} b^{2}\right )} x^{2} + {\left (8 i \, \pi ^{3} b - 48 \, \pi ^{2} a b - 96 i \, \pi a^{2} b + 64 \, a^{3} b\right )} x} + \frac {8 \, \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} - \frac {8 \, \log \relax (x)}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.49, size = 902, normalized size = 9.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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