Optimal. Leaf size=49 \[ -\frac{i \sqrt{\frac{b \cosh ^2(x)}{a}+1} \text{EllipticF}\left (\frac{\pi }{2}+i x,-\frac{b}{a}\right )}{\sqrt{a+b \cosh ^2(x)}} \]
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Rubi [A] time = 0.0297244, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3183, 3182} \[ -\frac{i \sqrt{\frac{b \cosh ^2(x)}{a}+1} F\left (i x+\frac{\pi }{2}|-\frac{b}{a}\right )}{\sqrt{a+b \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \cosh ^2(x)}} \, dx &=\frac{\sqrt{1+\frac{b \cosh ^2(x)}{a}} \int \frac{1}{\sqrt{1+\frac{b \cosh ^2(x)}{a}}} \, dx}{\sqrt{a+b \cosh ^2(x)}}\\ &=-\frac{i \sqrt{1+\frac{b \cosh ^2(x)}{a}} F\left (\frac{\pi }{2}+i x|-\frac{b}{a}\right )}{\sqrt{a+b \cosh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0696823, size = 53, normalized size = 1.08 \[ -\frac{i \sqrt{\frac{2 a+b \cosh (2 x)+b}{a+b}} \text{EllipticF}\left (i x,\frac{b}{a+b}\right )}{\sqrt{2 a+b \cosh (2 x)+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 66, normalized size = 1.4 \begin{align*}{\frac{1}{\sinh \left ( x \right ) }\sqrt{{\frac{a+b \left ( \cosh \left ( x \right ) \right ) ^{2}}{a}}}\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x \right ) \sqrt{-{\frac{b}{a}}},\sqrt{-{\frac{a}{b}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}{\frac{1}{\sqrt{a+b \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \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{1}{\sqrt{b \cosh \left (x\right )^{2} + 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{1}{\sqrt{b \cosh \left (x\right )^{2} + 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{1}{\sqrt{a + b \cosh ^{2}{\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{1}{\sqrt{b \cosh \left (x\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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