Integrand size = 13, antiderivative size = 103 \[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\frac {2}{5} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^{3/2}-\frac {6 i \left (a^2-b^2\right ) E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{5 \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Time = 0.04 (sec) , antiderivative size = 103, 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 = {3152, 3157, 2719} \[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\frac {2}{5} (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))^{3/2}-\frac {6 i \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{5 \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Rule 2719
Rule 3152
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^{3/2}+\frac {1}{5} \left (3 \left (a^2-b^2\right )\right ) \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx \\ & = \frac {2}{5} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^{3/2}+\frac {\left (3 \left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}\right ) \int \sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{5 \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ & = \frac {2}{5} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^{3/2}-\frac {6 i \left (a^2-b^2\right ) E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{5 \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.59 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.87 \[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\frac {(a \cosh (x)+b \sinh (x)) \left (6 a \left (a^2-b^2\right )+2 a b^2 \cosh (2 x)+b \left (a^2+b^2\right ) \sinh (2 x)\right )-\frac {3 (a-b)^2 (a+b)^2 \left (b \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cosh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right ) \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )+\sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )} \left (2 a \cosh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )-b \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right )\right )}{a \sqrt {1-\frac {b^2}{a^2}} \sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )}}}{5 b \sqrt {a \cosh (x)+b \sinh (x)}} \]
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Time = 0.91 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.41
method | result | size |
default | \(-\frac {\left (\left (a +b \right ) \left (a -b \right )\right )^{\frac {3}{2}} \left (\frac {\cosh \left (x \right )^{3}}{3}-\cosh \left (x \right )\right )}{\sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(42\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.66 \[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=-\frac {12 \, {\left (\sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {a + b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} - 12 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, a^{2} + 2 \, b^{2}\right )} \sinh \left (x\right )^{2} - a^{2} + 2 \, a b - b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} - 6 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}}{10 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \]
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Timed out. \[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\int { {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\int { {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx=\int {\left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \]
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