Integrand size = 18, antiderivative size = 42 \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \text {csch}^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b^3} \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5527, 4269, 3556} \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\frac {\log (\sinh (a+b x))}{b^3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \text {csch}^2(a+b x)}{2 b} \]
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Rule 3556
Rule 4269
Rule 5527
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \text {csch}^2(a+b x)}{2 b}+\frac {\int x \text {csch}^2(a+b x) \, dx}{b} \\ & = -\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \text {csch}^2(a+b x)}{2 b}+\frac {\int \coth (a+b x) \, dx}{b^2} \\ & = -\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \text {csch}^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31 \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {x \coth (a)}{b^2}-\frac {x^2 \text {csch}^2(a+b x)}{2 b}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2} \]
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Time = 0.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.71
method | result | size |
risch | \(-\frac {2 x}{b^{2}}-\frac {2 a}{b^{3}}-\frac {2 x \left ({\mathrm e}^{2 b x +2 a} b x +{\mathrm e}^{2 b x +2 a}-1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b^{3}}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (40) = 80\).
Time = 0.26 (sec) , antiderivative size = 383, normalized size of antiderivative = 9.12 \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, b x \cosh \left (b x + a\right )^{4} + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 2 \, b x \sinh \left (b x + a\right )^{4} + 2 \, {\left (b^{2} x^{2} - b x\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} + 6 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \, {\left (2 \, b x \cosh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b^{3} \cosh \left (b x + a\right )^{4} + 4 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{3} \sinh \left (b x + a\right )^{4} - 2 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} - b^{3}\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} - b^{3} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )} \]
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\[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\int x^{2} \cosh {\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (40) = 80\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.55 \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, {\left ({\left (b x^{2} e^{\left (2 \, a\right )} - x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + x e^{\left (4 \, b x + 4 \, a\right )}\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{3}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (40) = 80\).
Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.31 \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b x e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b x e^{\left (2 \, b x + 2 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) - \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{b^{3} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{3}} \]
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Time = 2.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.40 \[ \int x^2 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b^3}-\frac {\frac {x^2}{b}+\frac {x^2\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b}}{{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1}-\frac {2\,x}{b^2}-\frac {b\,x^2+2\,x}{b^2\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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