Integrand size = 15, antiderivative size = 15 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2686, 30} \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b} \]
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Rule 30
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int x \, dx,x,-i \text {csch}(a+b x))}{b} \\ & = -\frac {\text {csch}^2(a+b x)}{2 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b} \]
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Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{2 b}\) | \(14\) |
default | \(-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{2 b}\) | \(14\) |
risch | \(-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\) | \(28\) |
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (13) = 26\).
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 5.73 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right ) + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )} \]
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\[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=\int \cosh {\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \]
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Time = 2.38 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {1}{2\,b\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \]
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