Integrand size = 11, antiderivative size = 60 \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2833, 12, 2739, 632, 212} \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}} \]
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Rule 12
Rule 212
Rule 632
Rule 2739
Rule 2833
Rubi steps \begin{align*} \text {integral}& = \frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {b}{a+b \sinh (x)} \, dx}{a^2+b^2} \\ & = \frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {b \int \frac {1}{a+b \sinh (x)} \, dx}{a^2+b^2} \\ & = \frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = \frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {(4 b) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2} \\ & = -\frac {2 b \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 b \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\frac {a \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Time = 0.45 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.62
method | result | size |
default | \(\frac {8 b \tanh \left (\frac {x}{2}\right )+8 a}{\left (-4 a^{2}-4 b^{2}\right ) \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}-\frac {8 b \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (-4 a^{2}-4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\) | \(97\) |
risch | \(-\frac {2 a \left ({\mathrm e}^{x} a -b \right )}{b \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(155\) |
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Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (56) = 112\).
Time = 0.26 (sec) , antiderivative size = 341, normalized size of antiderivative = 5.68 \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, a^{3} b + 2 \, a b^{3} + {\left (b^{3} \cosh \left (x\right )^{2} + b^{3} \sinh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) - b^{3} + 2 \, {\left (b^{3} \cosh \left (x\right ) + a b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} \sinh \left (x\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )^{2} - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
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Timed out. \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (56) = 112\).
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.95 \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (a^{2} e^{\left (-x\right )} + a b\right )}}{a^{2} b^{2} + b^{4} + 2 \, {\left (a^{3} b + a b^{3}\right )} e^{\left (-x\right )} - {\left (a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.65 \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {b \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a^{2} e^{x} - a b\right )}}{{\left (a^{2} b + b^{3}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} \]
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Time = 1.41 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.37 \[ \int \frac {\sinh (x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2\,a\,b}{a^2\,b+b^3}-\frac {2\,a^2\,{\mathrm {e}}^x}{a^2\,b+b^3}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {b\,\ln \left (-\frac {2\,{\mathrm {e}}^x}{a^2+b^2}-\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}}+\frac {b\,\ln \left (\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^x}{a^2+b^2}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]
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