Integrand size = 13, antiderivative size = 38 \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{b^3}-\frac {a \sinh (x)}{b^2}+\frac {\sinh ^2(x)}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2747, 711} \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=\frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{b^3}-\frac {a \sinh (x)}{b^2}+\frac {\sinh ^2(x)}{2 b} \]
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Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {-b^2-x^2}{a+x} \, dx,x,b \sinh (x)\right )}{b^3} \\ & = -\frac {\text {Subst}\left (\int \left (a-x+\frac {-a^2-b^2}{a+x}\right ) \, dx,x,b \sinh (x)\right )}{b^3} \\ & = \frac {\left (a^2+b^2\right ) \log (a+b \sinh (x))}{b^3}-\frac {a \sinh (x)}{b^2}+\frac {\sinh ^2(x)}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=-\frac {-\left (\left (a^2+b^2\right ) \log (a+b \sinh (x))\right )+a b \sinh (x)-\frac {1}{2} b^2 \sinh ^2(x)}{b^3} \]
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Time = 2.99 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-\frac {-\frac {b \sinh \left (x \right )^{2}}{2}+a \sinh \left (x \right )}{b^{2}}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (a +b \sinh \left (x \right )\right )}{b^{3}}\) | \(37\) |
default | \(-\frac {-\frac {b \sinh \left (x \right )^{2}}{2}+a \sinh \left (x \right )}{b^{2}}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (a +b \sinh \left (x \right )\right )}{b^{3}}\) | \(37\) |
risch | \(-\frac {x \,a^{2}}{b^{3}}-\frac {x}{b}+\frac {{\mathrm e}^{2 x}}{8 b}-\frac {a \,{\mathrm e}^{x}}{2 b^{2}}+\frac {a \,{\mathrm e}^{-x}}{2 b^{2}}+\frac {{\mathrm e}^{-2 x}}{8 b}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b}\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (36) = 72\).
Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 5.82 \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=\frac {b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} - 4 \, a b \cosh \left (x\right )^{3} - 8 \, {\left (a^{2} + b^{2}\right )} x \cosh \left (x\right )^{2} + 4 \, {\left (b^{2} \cosh \left (x\right ) - a b\right )} \sinh \left (x\right )^{3} + 4 \, a b \cosh \left (x\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} - 6 \, a b \cosh \left (x\right ) - 4 \, {\left (a^{2} + b^{2}\right )} x\right )} \sinh \left (x\right )^{2} + b^{2} + 8 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} - 3 \, a b \cosh \left (x\right )^{2} - 4 \, {\left (a^{2} + b^{2}\right )} x \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{8 \, {\left (b^{3} \cosh \left (x\right )^{2} + 2 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right ) + b^{3} \sinh \left (x\right )^{2}\right )}} \]
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Timed out. \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (36) = 72\).
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.13 \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (4 \, a e^{\left (-x\right )} - b\right )} e^{\left (2 \, x\right )}}{8 \, b^{2}} + \frac {4 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )}}{8 \, b^{2}} + \frac {{\left (a^{2} + b^{2}\right )} x}{b^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=\frac {b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, a {\left (e^{\left (-x\right )} - e^{x}\right )}}{8 \, b^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{3}} \]
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Time = 1.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.03 \[ \int \frac {\cosh ^3(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,x}}{8\,b}+\frac {\ln \left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^2+b^2\right )}{b^3}-\frac {a\,{\mathrm {e}}^x}{2\,b^2}-\frac {x\,\left (a^2+b^2\right )}{b^3}+\frac {a\,{\mathrm {e}}^{-x}}{2\,b^2} \]
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