Integrand size = 8, antiderivative size = 11 \[ \int \text {csch}^2(a+b x) \, dx=-\frac {\coth (a+b x)}{b} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852, 8} \[ \int \text {csch}^2(a+b x) \, dx=-\frac {\coth (a+b x)}{b} \]
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Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}(\int 1 \, dx,x,-i \coth (a+b x))}{b} \\ & = -\frac {\coth (a+b x)}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \text {csch}^2(a+b x) \, dx=-\frac {\coth (a+b x)}{b} \]
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Time = 0.76 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(-\frac {\coth \left (b x +a \right )}{b}\) | \(12\) |
default | \(-\frac {\coth \left (b x +a \right )}{b}\) | \(12\) |
risch | \(-\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}\) | \(19\) |
parallelrisch | \(\frac {-\coth \left (\frac {b x}{2}+\frac {a}{2}\right )-\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (11) = 22\).
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 3.91 \[ \int \text {csch}^2(a+b x) \, dx=-\frac {2}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
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\[ \int \text {csch}^2(a+b x) \, dx=\int \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.64 \[ \int \text {csch}^2(a+b x) \, dx=\frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.64 \[ \int \text {csch}^2(a+b x) \, dx=-\frac {2}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.64 \[ \int \text {csch}^2(a+b x) \, dx=-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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