Integrand size = 11, antiderivative size = 57 \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=-\frac {b x}{a^2}-\frac {2 b^2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {\cosh (x)}{a} \]
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Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3938, 12, 3868, 2739, 632, 212} \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=-\frac {2 b^2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}-\frac {b x}{a^2}+\frac {\cosh (x)}{a} \]
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Rule 12
Rule 212
Rule 632
Rule 2739
Rule 3868
Rule 3938
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (x)}{a}-\frac {\int \frac {b}{a+b \text {csch}(x)} \, dx}{a} \\ & = \frac {\cosh (x)}{a}-\frac {b \int \frac {1}{a+b \text {csch}(x)} \, dx}{a} \\ & = -\frac {b x}{a^2}+\frac {\cosh (x)}{a}+\frac {b \int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx}{a^2} \\ & = -\frac {b x}{a^2}+\frac {\cosh (x)}{a}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2} \\ & = -\frac {b x}{a^2}+\frac {\cosh (x)}{a}-\frac {(4 b) \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 )}{a^2} \\ & = -\frac {b x}{a^2}-\frac {2 b^2 \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2}}+\frac {\cosh (x)}{a} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=\frac {b \left (-x+\frac {2 b \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}\right )+a \cosh (x)}{a^2} \]
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Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.61
method | result | size |
default | \(\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{2}}-\frac {2 b^{2} \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} \sqrt {a^{2}+b^{2}}}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{2}}\) | \(92\) |
risch | \(-\frac {b x}{a^{2}}+\frac {{\mathrm e}^{x}}{2 a}+\frac {{\mathrm e}^{-x}}{2 a}+\frac {b^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}\, a^{2}}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (53) = 106\).
Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 4.18 \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=\frac {a^{3} + a b^{2} - 2 \, {\left (a^{2} b + b^{3}\right )} x \cosh \left (x\right ) + {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} + {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) - 2 \, {\left ({\left (a^{2} b + b^{3}\right )} x - {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left ({\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right ) + {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (x\right )\right )}} \]
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\[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=\int \frac {\sinh {\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47 \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=\frac {b^{2} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{2}} - \frac {b x}{a^{2}} + \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} \]
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Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.51 \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=\frac {b^{2} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{2}} - \frac {b x}{a^{2}} + \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} \]
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Time = 2.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.26 \[ \int \frac {\sinh (x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,a}+\frac {{\mathrm {e}}^x}{2\,a}-\frac {b\,x}{a^2}-\frac {b^2\,\ln \left (-\frac {2\,b^2\,{\mathrm {e}}^x}{a^3}-\frac {2\,b^2\,\left (a-b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a^2+b^2}}\right )}{a^2\,\sqrt {a^2+b^2}}+\frac {b^2\,\ln \left (\frac {2\,b^2\,\left (a-b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a^2+b^2}}-\frac {2\,b^2\,{\mathrm {e}}^x}{a^3}\right )}{a^2\,\sqrt {a^2+b^2}} \]
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