Integrand size = 11, antiderivative size = 67 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=\frac {b \cosh (x)+a \sinh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3}+\frac {2 \sinh (x)}{3 a \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3155, 3154} \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=\frac {2 \sinh (x)}{3 a \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac {a \sinh (x)+b \cosh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3} \]
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Rule 3154
Rule 3155
Rubi steps \begin{align*} \text {integral}& = \frac {b \cosh (x)+a \sinh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3}+\frac {2 \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx}{3 \left (a^2-b^2\right )} \\ & = \frac {b \cosh (x)+a \sinh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3}+\frac {2 \sinh (x)}{3 a \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=\frac {a b \cosh (3 x)+\left (2 a^2-b^2+\left (a^2+b^2\right ) \cosh (2 x)\right ) \sinh (x)}{3 a (a-b) (a+b) (a \cosh (x)+b \sinh (x))^3} \]
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Time = 137.48 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {4 \left (3 a \,{\mathrm e}^{2 x}+3 b \,{\mathrm e}^{2 x}+a -b \right )}{3 \left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{2 x}+a -b \right )^{3} \left (a +b \right )^{2}}\) | \(46\) |
default | \(-\frac {2 \left (-\frac {\tanh \left (\frac {x}{2}\right )^{5}}{a}-\frac {2 b \tanh \left (\frac {x}{2}\right )^{4}}{a^{2}}-\frac {2 \left (a^{2}+2 b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{3 a^{3}}-\frac {2 b \tanh \left (\frac {x}{2}\right )^{2}}{a^{2}}-\frac {\tanh \left (\frac {x}{2}\right )}{a}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )^{3}}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (63) = 126\).
Time = 0.26 (sec) , antiderivative size = 527, normalized size of antiderivative = 7.87 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=-\frac {8 \, {\left ({\left (2 \, a + b\right )} \cosh \left (x\right ) + {\left (a + 2 \, b\right )} \sinh \left (x\right )\right )}}{3 \, {\left ({\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{5} + 5 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \sinh \left (x\right )^{5} + 3 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{3} + {\left (3 \, a^{5} + 9 \, a^{4} b + 6 \, a^{3} b^{2} - 6 \, a^{2} b^{3} - 9 \, a b^{4} - 3 \, b^{5} + 10 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} + {\left (10 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{3} + 9 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, a^{5} + a^{4} b - 4 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} \cosh \left (x\right ) + {\left (2 \, a^{5} + 4 \, a^{4} b - 4 \, a^{3} b^{2} - 8 \, a^{2} b^{3} + 2 \, a b^{4} + 4 \, b^{5} + 5 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{4} + 9 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (63) = 126\).
Time = 0.22 (sec) , antiderivative size = 498, normalized size of antiderivative = 7.43 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=\frac {4 \, {\left (a - b\right )} e^{\left (-2 \, x\right )}}{a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} e^{\left (-4 \, x\right )} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, x\right )}} + \frac {4 \, a}{3 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} e^{\left (-4 \, x\right )} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {4 \, b}{3 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} e^{\left (-4 \, x\right )} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, x\right )}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=-\frac {4 \, {\left (3 \, a e^{\left (2 \, x\right )} + 3 \, b e^{\left (2 \, x\right )} + a - b\right )}}{3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{3}} \]
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Time = 2.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^4} \, dx=-\frac {a\,\left (4\,{\mathrm {e}}^{2\,x}+\frac {4}{3}\right )+b\,\left (4\,{\mathrm {e}}^{2\,x}-\frac {4}{3}\right )}{{\left (a+b\right )}^2\,{\left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}^3} \]
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