Integrand size = 18, antiderivative size = 30 \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {2 x \coth (2 a+2 b x)}{b}+\frac {\log (\sinh (2 a+2 b x))}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5569, 4269, 3556} \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\frac {\log (\sinh (2 a+2 b x))}{b^2}-\frac {2 x \coth (2 a+2 b x)}{b} \]
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Rule 3556
Rule 4269
Rule 5569
Rubi steps \begin{align*} \text {integral}& = 4 \int x \text {csch}^2(2 a+2 b x) \, dx \\ & = -\frac {2 x \coth (2 a+2 b x)}{b}+\frac {2 \int \coth (2 a+2 b x) \, dx}{b} \\ & = -\frac {2 x \coth (2 a+2 b x)}{b}+\frac {\log (\sinh (2 a+2 b x))}{b^2} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\frac {-2 b x \coth (2 (a+b x))+\log (\sinh (2 (a+b x)))}{b^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(30)=60\).
Time = 3.53 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07
method | result | size |
risch | \(-\frac {4 x}{b}-\frac {4 a}{b^{2}}-\frac {4 x}{b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{4 b x +4 a}-1\right )}{b^{2}}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 292, normalized size of antiderivative = 9.73 \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 \, b x \cosh \left (b x + a\right )^{4} + 16 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 24 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, b x \sinh \left (b x + a\right )^{4} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \log \left (\frac {4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}}\right )}{b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b^{2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} - b^{2}} \]
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\[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int x \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (30) = 60\).
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90 \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 \, x e^{\left (4 \, b x + 4 \, a\right )}}{b e^{\left (4 \, b x + 4 \, a\right )} - b} + \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40 \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 \, b x e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right ) + \log \left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - b^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int x \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\frac {\ln \left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,b\,x}-1\right )}{b^2}-\frac {4\,x}{b}-\frac {4\,x}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )} \]
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