Optimal. Leaf size=108 \[ -\frac{2 i (A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}-\frac{2 i B \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]
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Rubi [A] time = 0.111749, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {2752, 2663, 2661, 2655, 2653} \[ -\frac{2 i (A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}-\frac{2 i B \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx &=\frac{B \int \sqrt{a+b \cosh (x)} \, dx}{b}+\frac{(A b-a B) \int \frac{1}{\sqrt{a+b \cosh (x)}} \, dx}{b}\\ &=\frac{\left (B \sqrt{a+b \cosh (x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{b \sqrt{\frac{a+b \cosh (x)}{a+b}}}+\frac{\left ((A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}}} \, dx}{b \sqrt{a+b \cosh (x)}}\\ &=-\frac{2 i B \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{2 i (A b-a B) \sqrt{\frac{a+b \cosh (x)}{a+b}} F\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.443158, size = 80, normalized size = 0.74 \[ -\frac{2 i \sqrt{\frac{a+b \cosh (x)}{a+b}} \left ((A b-a B) \text{EllipticF}\left (\frac{i x}{2},\frac{2 b}{a+b}\right )+B (a+b) E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )\right )}{b \sqrt{a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.096, size = 218, normalized size = 2. \begin{align*} 2\,{\frac{\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sqrt{ \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}b+a-b \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}}{\sqrt{2\,b \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sinh \left ( x/2 \right ) \sqrt{2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b+a+b}} \left ( A{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) +B{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) -2\,B{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) \right ) \sqrt{{\frac{2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}b+a-b}{a-b}}}{\frac{1}{\sqrt{-2\,{\frac{b}{a-b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{\sqrt{b \cosh \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B \cosh \left (x\right ) + A}{\sqrt{b \cosh \left (x\right ) + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B \cosh{\left (x \right )}}{\sqrt{a + b \cosh{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{\sqrt{b \cosh \left (x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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