Optimal. Leaf size=84 \[ -\frac{2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]
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Rubi [A] time = 0.0570477, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2664, 21, 2655, 2653} \[ -\frac{2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 i \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 21
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{1}{(a+b \cosh (x))^{3/2}} \, dx &=-\frac{2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}-\frac{2 \int \frac{-\frac{a}{2}-\frac{1}{2} b \cosh (x)}{\sqrt{a+b \cosh (x)}} \, dx}{a^2-b^2}\\ &=-\frac{2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}+\frac{\int \sqrt{a+b \cosh (x)} \, dx}{a^2-b^2}\\ &=-\frac{2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}+\frac{\sqrt{a+b \cosh (x)} \int \sqrt{\frac{a}{a+b}+\frac{b \cosh (x)}{a+b}} \, dx}{\left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}\\ &=-\frac{2 i \sqrt{a+b \cosh (x)} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt{\frac{a+b \cosh (x)}{a+b}}}-\frac{2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt{a+b \cosh (x)}}\\ \end{align*}
Mathematica [A] time = 0.120197, size = 68, normalized size = 0.81 \[ -\frac{2 \left (b \sinh (x)+i (a+b) \sqrt{\frac{a+b \cosh (x)}{a+b}} E\left (\frac{i x}{2}|\frac{2 b}{a+b}\right )\right )}{(a-b) (a+b) \sqrt{a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 296, normalized size = 3.5 \begin{align*} 2\,{\frac{1}{ \left ( a+b \right ) \left ( a-b \right ) \sinh \left ( x/2 \right ) \sqrt{2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b+a+b}} \left ( -2\,\sqrt{-2\,{\frac{b}{a-b}}}b\cosh \left ( x/2 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}+\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) a+\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) b-2\,\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}\sqrt{2\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}b}{a-b}}+{\frac{a+b}{a-b}}}{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{-2\,{\frac{b}{a-b}}},1/2\,\sqrt{-2\,{\frac{a-b}{b}}} \right ) b \right ){\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{1}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{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{\sqrt{b \cosh \left (x\right ) + a}}{b^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2}}, 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}{\left (a + b \cosh{\left (x \right )}\right )^{\frac{3}{2}}}\, 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}{{\left (b \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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