Integrand size = 11, antiderivative size = 37 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3916, 2739, 632, 212} \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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Rule 212
Rule 632
Rule 2739
Rule 3916
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b} \\ & = -\frac {4 \text {Subst}\left (\int \frac {1}{4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{b} \\ & = -\frac {2 \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}} \]
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Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\) | \(35\) |
risch | \(\frac {\ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}}\) | \(97\) |
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (33) = 66\).
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.00 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right )}{\sqrt {a^{2} + b^{2}}} \]
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\[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {csch}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.51 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} \]
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Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {-a^2-b^2}}+\frac {a\,{\mathrm {e}}^x}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}} \]
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