Optimal. Leaf size=65 \[ -\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} F\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {a \cosh (x)+b \sinh (x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3078, 2641} \[ -\frac {2 i \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} F\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {a \cosh (x)+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3078
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx &=\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 F\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\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*}
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Mathematica [C] time = 0.10, size = 81, normalized size = 1.25 \[ \frac {2 \sqrt {a \cosh (x)+b \sinh (x)} \sqrt {\cosh ^2\left (\tanh ^{-1}\left (\frac {a}{b}\right )+x\right )} \text {sech}\left (\tanh ^{-1}\left (\frac {a}{b}\right )+x\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\tanh ^{-1}\left (\frac {a}{b}\right )\right )\right )}{b \sqrt {1-\frac {a^2}{b^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {a \cosh \relax (x) + b \sinh \relax (x)}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh \relax (x) + b \sinh \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 97, normalized size = 1.49 \[ \frac {\sqrt {-\sqrt {a^{2}-b^{2}}\, \left (\sinh ^{3}\relax (x )\right )}\, \arctan \left (\frac {\sqrt {\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}\, \cosh \relax (x )}{\sqrt {-\sqrt {a^{2}-b^{2}}\, \left (\sinh ^{3}\relax (x )\right )}}\right )}{\sqrt {\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}\, \sinh \relax (x ) \sqrt {-\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh \relax (x) + b \sinh \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a\,\mathrm {cosh}\relax (x)+b\,\mathrm {sinh}\relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cosh {\relax (x )} + b \sinh {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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