Integrand size = 13, antiderivative size = 112 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {2 i E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Time = 0.04 (sec) , antiderivative size = 112, 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, 2719} \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\frac {2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Rule 2719
Rule 3155
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {\int \sqrt {a \cosh (x)+b \sinh (x)} \, dx}{-a^2+b^2} \\ & = \frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {\sqrt {a \cosh (x)+b \sinh (x)} \int \sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\left (-a^2+b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \\ & = \frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {2 i 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)}}{\left (a^2-b^2\right ) \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 = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\frac {b \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cosh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right ) \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )-\sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )} \left (2 a \sqrt {1-\frac {b^2}{a^2}} \cosh (x)-2 a \cosh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )+b \sinh \left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )\right )}{a b \sqrt {1-\frac {b^2}{a^2}} \sqrt {a \cosh (x)+b \sinh (x)} \sqrt {-\sinh ^2\left (x+\text {arctanh}\left (\frac {b}{a}\right )\right )}} \]
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Time = 0.59 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\cosh \left (x \right )\right )}{\sqrt {a^{2}-b^{2}}\, \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(33\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} {\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a + b\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (a - b\right )}\right )} \sqrt {a + b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, {\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2}\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}\right )}}{a^{3} - a^{2} b - a b^{2} + b^{3} + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \sinh \left (x\right )^{2}} \]
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\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\int \frac {1}{\left (a \cosh {\left (x \right )} + b \sinh {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx=\int \frac {1}{{\left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2}} \,d x \]
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