Optimal. Leaf size=65 \[ -\frac {2 i F\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{\sqrt {a \cosh (x)+b \sinh (x)}} \]
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Rubi [A]
time = 0.02, 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 = {3157, 2720}
\begin {gather*} -\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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3157
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.07, size = 81, normalized size = 1.25 \begin {gather*} \frac {2 \sqrt {\cosh ^2\left (x+\tanh ^{-1}\left (\frac {a}{b}\right )\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 ) \text {sech}\left (x+\tanh ^{-1}\left (\frac {a}{b}\right )\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\sqrt {1-\frac {a^2}{b^2}} b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.14, size = 97, normalized size = 1.49
method | result | size |
default | \(\frac {\sqrt {-\sqrt {a^{2}-b^{2}}\, \left (\sinh ^{3}\left (x \right )\right )}\, \arctan \left (\frac {\sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \cosh \left (x \right )}{\sqrt {-\sqrt {a^{2}-b^{2}}\, \left (\sinh ^{3}\left (x \right )\right )}}\right )}{\sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \sinh \left (x \right ) \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\) | \(97\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 29, normalized size = 0.45 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a \cosh {\left (x \right )} + b \sinh {\left (x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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