Optimal. Leaf size=65 \[ -\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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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, 2639} \[ -\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3078
Rubi steps
\begin {align*} \int \sqrt {a \cosh (x)+b \sinh (x)} \, dx &=\frac {\sqrt {a \cosh (x)+b \sinh (x)} \int \sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}\\ &=-\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)}}{\sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}\\ \end {align*}
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Mathematica [C] time = 0.76, size = 206, normalized size = 3.17 \[ \frac {b \left (b^2-a^2\right ) \sinh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cosh ^2\left (x+\tanh ^{-1}\left (\frac {b}{a}\right )\right )\right )+\sqrt {-\sinh ^2\left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )} \left (2 a^2 b \sqrt {1-\frac {b^2}{a^2}} \sinh (x)-2 a \left (a^2-b^2\right ) \cosh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )+a^2 b \sinh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )+2 a^3 \sqrt {1-\frac {b^2}{a^2}} \cosh (x)-b^3 \sinh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )\right )}{a b \sqrt {1-\frac {b^2}{a^2}} \sqrt {-\sinh ^2\left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )} \sqrt {a \cosh (x)+b \sinh (x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x) + b \sinh \relax (x)}, x\right ) \]
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 \[ \int \sqrt {a \cosh \relax (x) + b \sinh \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 33, normalized size = 0.51 \[ -\frac {\sqrt {a^{2}-b^{2}}\, \cosh \relax (x )}{\sqrt {-\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}} \]
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 \[ \int \sqrt {a \cosh \relax (x) + b \sinh \relax (x)}\,{d x} \]
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 \[ \int \sqrt {a\,\mathrm {cosh}\relax (x)+b\,\mathrm {sinh}\relax (x)} \,d x \]
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 \[ \int \sqrt {a \cosh {\relax (x )} + b \sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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