Integrand size = 13, antiderivative size = 60 \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=\frac {2 a b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2} \]
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Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3957, 2945, 12, 2739, 632, 210} \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=\frac {2 a b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2} \]
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2945
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\text {sech}(x) \tanh (x)}{i b+i a \sinh (x)} \, dx \\ & = -\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2}-\frac {i \int \frac {a b}{i b+i a \sinh (x)} \, dx}{a^2+b^2} \\ & = -\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2}-\frac {(i a b) \int \frac {1}{i b+i a \sinh (x)} \, dx}{a^2+b^2} \\ & = -\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2}-\frac {(2 i a b) \text {Subst}\left (\int \frac {1}{i b+2 i a x-i b x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = -\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2}+\frac {(4 i a b) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,2 i a-2 i b \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = \frac {2 a b \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {\text {sech}(x) (b-a \sinh (x))}{a^2+b^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12 \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=\frac {-b \text {sech}(x)+a \left (-\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\tanh (x)\right )}{a^2+b^2} \]
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Time = 0.49 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {4 a b \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}-\frac {2 \left (-a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}\) | \(81\) |
risch | \(-\frac {2 \left ({\mathrm e}^{x} b +a \right )}{\left (1+{\mathrm e}^{2 x}\right ) \left (a^{2}+b^{2}\right )}+\frac {b a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {b a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(142\) |
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Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (56) = 112\).
Time = 0.24 (sec) , antiderivative size = 256, normalized size of antiderivative = 4.27 \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 \, a^{3} + 2 \, a b^{2} - {\left (a b \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) \sinh \left (x\right ) + a b \sinh \left (x\right )^{2} + a b\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) + 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{2} b + b^{3}\right )} \sinh \left (x\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}} \]
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\[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {sech}^{2}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.52 \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {a b \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{\left (-x\right )} - a\right )}}{a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.42 \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=-\frac {a b \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
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Time = 2.37 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.22 \[ \int \frac {\text {sech}^2(x)}{a+b \text {csch}(x)} \, dx=\frac {a\,b\,\ln \left (\frac {2\,b\,\left (a-b\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{3/2}}+\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {a\,b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}-\frac {2\,b\,\left (a-b\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {\frac {2\,a}{a^2+b^2}+\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}}{{\mathrm {e}}^{2\,x}+1} \]
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