Integrand size = 13, antiderivative size = 116 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\frac {2 (b \cosh (x)+a \sinh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}} \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3155, 3157, 2720} \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\frac {2 (a \sinh (x)+b \cosh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}-\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}} \]
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Rule 2720
Rule 3155
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b \cosh (x)+a \sinh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx}{3 \left (a^2-b^2\right )} \\ & = \frac {2 (b \cosh (x)+a \sinh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}+\frac {\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \int \frac {1}{\sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )}} \, dx}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}} \\ & = \frac {2 (b \cosh (x)+a \sinh (x))}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^{3/2}}-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.43 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=-\frac {2 \left (\sqrt {1-\frac {a^2}{b^2}} b (b \cosh (x)+a \sinh (x))+\sqrt {\cosh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right ) (a \cosh (x)+b \sinh (x))^2\right )}{3 \sqrt {1-\frac {a^2}{b^2}} b (-a+b) (a+b) (a \cosh (x)+b \sinh (x))^{3/2}} \]
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Time = 0.63 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.32
method | result | size |
default | \(-\frac {\cosh \left (x \right )}{\left (a^{2}-b^{2}\right ) \sinh \left (x \right ) \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(37\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 679, normalized size of antiderivative = 5.85 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (a^{2} - b^{2}\right )}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b + b^{2}\right )}\right )} \sqrt {a + b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{3} - {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) + {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}\right )}}{3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\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 )^{4} + 4 \, {\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 ) \sinh \left (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 )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2} + 2 \, {\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} + 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 )^{2} + 4 \, {\left ({\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 (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx=\int \frac {1}{{\left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2}} \,d x \]
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