Integrand size = 17, antiderivative size = 25 \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=\frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5755, 2747, 31} \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=\frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \]
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Rule 31
Rule 2747
Rule 5755
Rubi steps \begin{align*} \text {integral}& = -\frac {x \cosh (x)}{a+b \sinh (x)}+\int \frac {\cosh (x)}{a+b \sinh (x)} \, dx \\ & = -\frac {x \cosh (x)}{a+b \sinh (x)}+\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{b} \\ & = \frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=\frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(25)=50\).
Time = 0.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32
method | result | size |
risch | \(-\frac {2 x}{b}+\frac {2 x \left (a \,{\mathrm e}^{x}-b \right )}{b \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b}\) | \(58\) |
parallelrisch | \(\frac {\left (a +b \sinh \left (x \right )\right ) \ln \left (\frac {-2 b \sinh \left (x \right )-2 a}{\cosh \left (x \right )+1}\right )+\left (-2 b \sinh \left (x \right )-2 a \right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )-x \left (a +b \cosh \left (x \right )+b \sinh \left (x \right )\right )}{b \left (a +b \sinh \left (x \right )\right )}\) | \(70\) |
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.36 \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, b x \cosh \left (x\right )^{2} + 2 \, b x \sinh \left (x\right )^{2} + 2 \, a x \cosh \left (x\right ) - {\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (2 \, b x \cosh \left (x\right ) + a x\right )} \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \]
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Timed out. \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.41 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48 \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, {\left (b x e^{\left (2 \, x\right )} + a x e^{x}\right )}}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} - b^{2}} + \frac {\log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b}{b}\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.84 \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, b x e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right ) - 2 \, a e^{x} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right ) + 2 \, b x + b \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} - b^{2}} \]
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Time = 2.43 (sec) , antiderivative size = 103, normalized size of antiderivative = 4.12 \[ \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx=\frac {\ln \left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )}{b}-\frac {\frac {2\,\left (x\,a^2\,b+x\,b^3\right )}{a^2\,b+b^3}-\frac {2\,{\mathrm {e}}^x\,\left (x\,a^3\,b+x\,a\,b^3\right )}{b\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {2\,x}{b} \]
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