Optimal. Leaf size=170 \[ \frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2} \]
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Rubi [A] time = 0.13, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2160
Rule 2163
Rule 2171
Rubi steps
\begin {align*} \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx &=\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {(2 b) \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))}\\ &=\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, 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 )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}+\frac {\left (6 b^2\right ) \int \frac {1}{x \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 )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {\left (6 b^2\right ) \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\left (6 b^3\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \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 )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {\left (6 b^2\right ) \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 )^3 \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 )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 107, normalized size = 0.63 \[ \frac {8 b^3 x^3 \coth ^{-1}(\tanh (a+b x))-12 b^2 x^2 \coth ^{-1}(\tanh (a+b x))^2 \left (\log (x)-\log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )-8 b x \coth ^{-1}(\tanh (a+b x))^3+\coth ^{-1}(\tanh (a+b x))^4-b^4 x^4}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5 \coth ^{-1}(\tanh (a+b x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1316, normalized size = 7.74 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^{3} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.15, size = 332, normalized size = 1.95 \[ \frac {192 \, b^{2} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} - \frac {192 \, b^{2} \log \relax (x)}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} + \frac {8 \, {\left (96 \, b^{3} x^{3} - i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3} + {\left (-72 i \, \pi b^{2} + 144 \, a b^{2}\right )} x^{2} - {\left (8 \, \pi ^{2} b + 32 i \, \pi a b - 32 \, a^{2} b\right )} x\right )}}{{\left (8 \, \pi ^{4} b^{2} + 64 i \, \pi ^{3} a b^{2} - 192 \, \pi ^{2} a^{2} b^{2} - 256 i \, \pi a^{3} b^{2} + 128 \, a^{4} b^{2}\right )} x^{4} + {\left (-8 i \, \pi ^{5} b + 80 \, \pi ^{4} a b + 320 i \, \pi ^{3} a^{2} b - 640 \, \pi ^{2} a^{3} b - 640 i \, \pi a^{4} b + 256 \, a^{5} b\right )} x^{3} - {\left (2 \, \pi ^{6} + 24 i \, \pi ^{5} a - 120 \, \pi ^{4} a^{2} - 320 i \, \pi ^{3} a^{3} + 480 \, \pi ^{2} a^{4} + 384 i \, \pi a^{5} - 128 \, a^{6}\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.12, size = 1251, normalized size = 7.36 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{x^{3} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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