Integrand size = 16, antiderivative size = 30 \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\coth (a+b x)}{2 b^2}-\frac {x \text {csch}^2(a+b x)}{2 b} \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5527, 3852, 8} \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\coth (a+b x)}{2 b^2}-\frac {x \text {csch}^2(a+b x)}{2 b} \]
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Rule 8
Rule 3852
Rule 5527
Rubi steps \begin{align*} \text {integral}& = -\frac {x \text {csch}^2(a+b x)}{2 b}+\frac {\int \text {csch}^2(a+b x) \, dx}{2 b} \\ & = -\frac {x \text {csch}^2(a+b x)}{2 b}-\frac {i \text {Subst}(\int 1 \, dx,x,-i \coth (a+b x))}{2 b^2} \\ & = -\frac {\coth (a+b x)}{2 b^2}-\frac {x \text {csch}^2(a+b x)}{2 b} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\coth (a+b x)}{2 b^2}-\frac {x \text {csch}^2(a+b x)}{2 b} \]
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Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{2 b x +2 a} b x +{\mathrm e}^{2 b x +2 a}-1}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\) | \(43\) |
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.57 \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, {\left (b x \cosh \left (b x + a\right ) + {\left (b x + 1\right )} \sinh \left (b x + a\right )\right )}}{b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} - b^{2} \cosh \left (b x + a\right ) + 3 \, {\left (b^{2} \cosh \left (b x + a\right )^{2} - b^{2}\right )} \sinh \left (b x + a\right )} \]
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\[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=\int x \cosh {\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.33 \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=\frac {2 \, b x e^{\left (4 \, b x + 4 \, a\right )} - {\left (4 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 1}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} - \frac {2 \, b x e^{\left (4 \, b x + 4 \, a\right )} + e^{\left (2 \, b x + 2 \, a\right )} - 1}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 6.13 \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {4 \, b x e^{\left (2 \, b x + 2 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + e^{\left (4 \, b x + 4 \, a\right )} \log \left (-e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (-e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} - \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + \log \left (-e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - 2}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} \]
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Time = 2.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int x \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (2\,b\,x+1\right )-1}{b^2\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^2} \]
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