Integrand size = 13, antiderivative size = 11 \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {csch}(a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2686, 8} \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {csch}(a+b x)}{b} \]
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Rule 8
Rule 2686
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}(\int 1 \, dx,x,-i \text {csch}(a+b x))}{b} \\ & = -\frac {\text {csch}(a+b x)}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {csch}(a+b x)}{b} \]
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Time = 0.16 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(-\frac {\operatorname {csch}\left (b x +a \right )}{b}\) | \(12\) |
default | \(-\frac {\operatorname {csch}\left (b x +a \right )}{b}\) | \(12\) |
risch | \(-\frac {2 \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}\) | \(25\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (11) = 22\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 5.09 \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
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\[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=\int \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \]
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Time = 2.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {1}{b\,\mathrm {sinh}\left (a+b\,x\right )} \]
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