Integrand size = 13, antiderivative size = 65 \[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{\sqrt {a \cosh (x)+b \sinh (x)}} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3157, 2720} \[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=-\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{\sqrt {a \cosh (x)+b \sinh (x)}} \]
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Rule 2720
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \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}{\sqrt {a \cosh (x)+b \sinh (x)}} \\ & = -\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}}}}{\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 = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=\frac {2 \sqrt {\cosh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\sqrt {1-\frac {a^2}{b^2}} b} \]
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Time = 0.65 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {\sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}\, \arctan \left (\frac {\sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \cosh \left (x \right )}{\sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}}\right )}{\sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \sinh \left (x \right ) \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(97\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=\frac {2 \, \sqrt {2} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )}{\sqrt {a + b}} \]
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\[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=\int \frac {1}{\sqrt {a \cosh {\left (x \right )} + b \sinh {\left (x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=\int { \frac {1}{\sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=\int { \frac {1}{\sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx=\int \frac {1}{\sqrt {a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )}} \,d x \]
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