Integrand size = 13, antiderivative size = 103 \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}-\frac {2 i \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \]
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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, 2720} \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2}{3} (a \sinh (x)+b \cosh (x)) \sqrt {a \cosh (x)+b \sinh (x)}-\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \]
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Rule 2720
Rule 3152
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}+\frac {1}{3} \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx \\ & = \frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}+\frac {\left (\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}\right ) \int \frac {1}{\sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )}} \, dx}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \\ & = \frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}-\frac {2 i \left (a^2-b^2\right ) \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 \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 = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2}{3} \left (b \cosh (x)-\sqrt {1-\frac {a^2}{b^2}} b \sqrt {\cosh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )+a \sinh (x)\right ) \sqrt {a \cosh (x)+b \sinh (x)} \]
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Time = 0.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {\sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}\, \left (\cosh \left (x \right ) \sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}\, \sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \left (a^{2}-b^{2}\right )+\sinh \left (x \right ) \left (a^{2}-b^{2}\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \cosh \left (x \right )}{\sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}}\right )\right )}{2 \sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \sinh \left (x \right )^{2} \sqrt {a^{2}-b^{2}}\, \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(171\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a - b\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a - b\right )} \sinh \left (x\right )\right )} \sqrt {a + b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - a + b\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}}{3 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
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\[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int \left (a \cosh {\left (x \right )} + b \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int { {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int { {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int {\left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \]
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