Integrand size = 14, antiderivative size = 25 \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5527, 3855} \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b} \]
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Rule 3855
Rule 5527
Rubi steps \begin{align*} \text {integral}& = -\frac {x \text {csch}(a+b x)}{b}+\frac {\int \text {csch}(a+b x) \, dx}{b} \\ & = -\frac {\text {arctanh}(\cosh (a+b x))}{b^2}-\frac {x \text {csch}(a+b x)}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(25)=50\).
Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.56 \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {x \text {csch}(a)}{b}-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b^2}+\frac {\log \left (\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b^2}+\frac {x \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b}+\frac {x \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(25)=50\).
Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16
method | result | size |
risch | \(-\frac {2 x \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 6.76 \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, b x \cosh \left (b x + a\right ) + 2 \, b x \sinh \left (b x + a\right ) + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}} \]
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\[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=\int x \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, x e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.72 \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, b x e^{\left (b x + a\right )} + e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) - e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - \log \left (e^{\left (b x + a\right )} + 1\right ) + \log \left (e^{\left (b x + a\right )} - 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} \]
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Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int x \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^4}}{b^2}\right )}{\sqrt {-b^4}}-\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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