Integrand size = 13, antiderivative size = 131 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \]
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Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2202, 2194, 2191, 2188, 29} \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {3 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4}+\frac {3 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4} \]
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Rule 29
Rule 2188
Rule 2191
Rule 2194
Rule 2202
Rubi steps \begin{align*} \text {integral}& = \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) \text {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*}
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {b^3 x^3-6 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+2 \coth ^{-1}(\tanh (a+b x))^3+3 b x \coth ^{-1}(\tanh (a+b x))^2 \left (1+2 \log (x)-2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 x \coth ^{-1}(\tanh (a+b x))^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^4} \]
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Timed out.
\[\int \frac {1}{x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}d x\]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {8 \, {\left (24 \, a b^{2} x^{2} + 36 \, a^{2} b x - i \, \pi ^{3} - 3 \, \pi ^{2} {\left (3 \, b x + 2 \, a\right )} + 8 \, a^{3} + 12 i \, \pi {\left (b^{2} x^{2} + 3 \, a b x + a^{2}\right )} - 6 \, {\left (4 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - \pi ^{2} b x + 4 \, a^{2} b x + 4 i \, \pi {\left (b^{2} x^{2} + a b x\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right ) + 6 \, {\left (4 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - \pi ^{2} b x + 4 \, a^{2} b x + 4 i \, \pi {\left (b^{2} x^{2} + a b x\right )}\right )} \log \left (x\right )\right )}}{64 \, a^{4} b^{2} x^{3} + 128 \, a^{5} b x^{2} - \pi ^{6} x + 64 \, a^{6} x + 4 i \, \pi ^{5} {\left (b x^{2} + 3 \, a x\right )} + 4 \, \pi ^{4} {\left (b^{2} x^{3} + 10 \, a b x^{2} + 15 \, a^{2} x\right )} - 32 i \, \pi ^{3} {\left (a b^{2} x^{3} + 5 \, a^{2} b x^{2} + 5 \, a^{3} x\right )} - 16 \, \pi ^{2} {\left (6 \, a^{2} b^{2} x^{3} + 20 \, a^{3} b x^{2} + 15 \, a^{4} x\right )} + 64 i \, \pi {\left (2 \, a^{3} b^{2} x^{3} + 5 \, a^{4} b x^{2} + 3 \, a^{5} x\right )}} \]
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\[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\int \frac {1}{x^{2} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.87 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {48 \, b \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {48 \, b \log \left (x\right )}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {8 \, {\left (12 \, b^{2} x^{2} - \pi ^{2} - 4 i \, \pi a + 4 \, a^{2} - 9 \, {\left (i \, \pi b - 2 \, a b\right )} x\right )}}{4 \, {\left (i \, \pi ^{3} b^{2} - 6 \, \pi ^{2} a b^{2} - 12 i \, \pi a^{2} b^{2} + 8 \, a^{3} b^{2}\right )} x^{3} + 4 \, {\left (\pi ^{4} b + 8 i \, \pi ^{3} a b - 24 \, \pi ^{2} a^{2} b - 32 i \, \pi a^{3} b + 16 \, a^{4} b\right )} x^{2} - {\left (i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}\right )} x} \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.97 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {48 \, b \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {48 \, b \log \left (x\right )}{\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {16 \, {\left (8 \, b^{2} x + 5 i \, \pi b + 10 \, a b\right )}}{8 i \, \pi ^{3} b^{2} x^{2} + 48 \, \pi ^{2} a b^{2} x^{2} - 96 i \, \pi a^{2} b^{2} x^{2} - 64 \, a^{3} b^{2} x^{2} - 8 \, \pi ^{4} b x + 64 i \, \pi ^{3} a b x + 192 \, \pi ^{2} a^{2} b x - 256 i \, \pi a^{3} b x - 128 \, a^{4} b x - 2 i \, \pi ^{5} - 20 \, \pi ^{4} a + 80 i \, \pi ^{3} a^{2} + 160 \, \pi ^{2} a^{3} - 160 i \, \pi a^{4} - 64 \, a^{5}} + \frac {8}{i \, \pi ^{3} x + 6 \, \pi ^{2} a x - 12 i \, \pi a^{2} x - 8 \, a^{3} x} \]
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Time = 8.29 (sec) , antiderivative size = 1074, normalized size of antiderivative = 8.20 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx=\text {Too large to display} \]
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