Integrand size = 11, antiderivative size = 47 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=\frac {x}{b}+\frac {2 a \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}} \]
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Time = 0.04 (sec) , antiderivative size = 47, 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 = {2814, 2739, 632, 212} \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=\frac {2 a \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}+\frac {x}{b} \]
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Rule 212
Rule 632
Rule 2739
Rule 2814
Rubi steps \begin{align*} \text {integral}& = \frac {x}{b}-\frac {a \int \frac {1}{a+b \sinh (x)} \, dx}{b} \\ & = \frac {x}{b}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b} \\ & = \frac {x}{b}+\frac {(4 a) \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 )}{b} \\ & = \frac {x}{b}+\frac {2 a \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=\frac {x-\frac {2 a \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}}{b} \]
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Time = 0.40 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.34
method | result | size |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}-\frac {2 a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \sqrt {a^{2}+b^{2}}}\) | \(63\) |
risch | \(\frac {x}{b}+\frac {a \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, b}-\frac {a \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, b}\) | \(110\) |
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.85 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} a \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) + {\left (a^{2} + b^{2}\right )} x}{a^{2} b + b^{3}} \]
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Result contains complex when optimal does not.
Time = 14.05 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.62 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{b} & \text {for}\: a = 0 \\\frac {\cosh {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {x \tanh {\left (\frac {x}{2} \right )}}{b \tanh {\left (\frac {x}{2} \right )} - i b} - \frac {i x}{b \tanh {\left (\frac {x}{2} \right )} - i b} - \frac {2}{b \tanh {\left (\frac {x}{2} \right )} - i b} & \text {for}\: a = - i b \\\frac {x \tanh {\left (\frac {x}{2} \right )}}{b \tanh {\left (\frac {x}{2} \right )} + i b} + \frac {i x}{b \tanh {\left (\frac {x}{2} \right )} + i b} - \frac {2}{b \tanh {\left (\frac {x}{2} \right )} + i b} & \text {for}\: a = i b \\\frac {a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} - \frac {a \log {\left (\tanh {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{b \sqrt {a^{2} + b^{2}}} + \frac {x}{b} & \text {otherwise} \end {cases} \]
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none
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.38 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=-\frac {a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b} + \frac {x}{b} \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=-\frac {a \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b} + \frac {x}{b} \]
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Time = 1.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.11 \[ \int \frac {\sinh (x)}{a+b \sinh (x)} \, dx=\frac {x}{b}-\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^2}-\frac {2\,a\,\left (b-a\,{\mathrm {e}}^x\right )}{b^2\,\sqrt {a^2+b^2}}\right )}{b\,\sqrt {a^2+b^2}}+\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^2}+\frac {2\,a\,\left (b-a\,{\mathrm {e}}^x\right )}{b^2\,\sqrt {a^2+b^2}}\right )}{b\,\sqrt {a^2+b^2}} \]
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