Integrand size = 14, antiderivative size = 29 \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=-\frac {(c+d x) \coth (a+b x)}{b}+\frac {d \log (\sinh (a+b x))}{b^2} \]
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Time = 0.02 (sec) , antiderivative size = 29, 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 = {4269, 3556} \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=\frac {d \log (\sinh (a+b x))}{b^2}-\frac {(c+d x) \coth (a+b x)}{b} \]
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Rule 3556
Rule 4269
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x) \coth (a+b x)}{b}+\frac {d \int \coth (a+b x) \, dx}{b} \\ & = -\frac {(c+d x) \coth (a+b x)}{b}+\frac {d \log (\sinh (a+b x))}{b^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=-\frac {d x \coth (a)}{b}-\frac {c \coth (a+b x)}{b}+\frac {d \log (\sinh (a+b x))}{b^2}+\frac {d x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b} \]
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Time = 0.80 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93
method | result | size |
risch | \(-\frac {2 d x}{b}-\frac {2 d a}{b^{2}}-\frac {2 \left (d x +c \right )}{\left ({\mathrm e}^{2 b x +2 a}-1\right ) b}+\frac {d \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b^{2}}\) | \(56\) |
parallelrisch | \(\frac {-4 \ln \left (1-\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +2 \ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -b \left (\coth \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (d x +c \right )+\left (d x +c \right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+2 d x \right )}{2 b^{2}}\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 5.72 \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, b d x \cosh \left (b x + a\right )^{2} + 4 \, b d x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, b d x \sinh \left (b x + a\right )^{2} + 2 \, b c - {\left (d \cosh \left (b x + a\right )^{2} + 2 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d \sinh \left (b x + a\right )^{2} - d\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}} \]
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\[ \int (c+d x) \text {csch}^2(a+b x) \, dx=\int \left (c + d x\right ) \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (29) = 58\).
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=-d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac {2 \, c}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (29) = 58\).
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.76 \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, b d x e^{\left (2 \, b x + 2 \, a\right )} - d e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, b c + d \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} \]
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Time = 2.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int (c+d x) \text {csch}^2(a+b x) \, dx=\frac {d\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b^2}-\frac {2\,\left (c+d\,x\right )}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2\,d\,x}{b} \]
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