Optimal. Leaf size=40 \[ \frac{2 a (3 A+B) \sinh (x)}{3 \sqrt{a \cosh (x)+a}}+\frac{2}{3} B \sinh (x) \sqrt{a \cosh (x)+a} \]
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Rubi [A] time = 0.0524759, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2751, 2646} \[ \frac{2 a (3 A+B) \sinh (x)}{3 \sqrt{a \cosh (x)+a}}+\frac{2}{3} B \sinh (x) \sqrt{a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \cosh (x)} (A+B \cosh (x)) \, dx &=\frac{2}{3} B \sqrt{a+a \cosh (x)} \sinh (x)+\frac{1}{3} (3 A+B) \int \sqrt{a+a \cosh (x)} \, dx\\ &=\frac{2 a (3 A+B) \sinh (x)}{3 \sqrt{a+a \cosh (x)}}+\frac{2}{3} B \sqrt{a+a \cosh (x)} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.038382, size = 31, normalized size = 0.78 \[ \frac{2}{3} \tanh \left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} (3 A+B \cosh (x)+2 B) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 39, normalized size = 1. \begin{align*}{\frac{2\,a\sqrt{2}}{3}\cosh \left ({\frac{x}{2}} \right ) \sinh \left ({\frac{x}{2}} \right ) \left ( 2\,B \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+3\,A+B \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6292, size = 122, normalized size = 3.05 \begin{align*}{\left (\sqrt{2} \sqrt{a} e^{\left (\frac{1}{2} \, x\right )} - \sqrt{2} \sqrt{a} e^{\left (-\frac{1}{2} \, x\right )}\right )} A + \frac{1}{6} \,{\left ({\left (\sqrt{2} \sqrt{a} e^{\left (-x\right )} + 3 \, \sqrt{2} \sqrt{a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac{5}{2} \, x\right )} -{\left (3 \, \sqrt{2} \sqrt{a} e^{\left (-x\right )} + \sqrt{2} \sqrt{a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac{1}{2} \, x\right )}\right )} B \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17391, size = 317, normalized size = 7.92 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (B \cosh \left (x\right )^{3} + B \sinh \left (x\right )^{3} + 3 \,{\left (2 \, A + B\right )} \cosh \left (x\right )^{2} + 3 \,{\left (B \cosh \left (x\right ) + 2 \, A + B\right )} \sinh \left (x\right )^{2} - 3 \,{\left (2 \, A + B\right )} \cosh \left (x\right ) + 3 \,{\left (B \cosh \left (x\right )^{2} + 2 \,{\left (2 \, A + B\right )} \cosh \left (x\right ) - 2 \, A - B\right )} \sinh \left (x\right ) - B\right )} \sqrt{\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{3 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\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 \sqrt{a \left (\cosh{\left (x \right )} + 1\right )} \left (A + B \cosh{\left (x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11449, size = 93, normalized size = 2.32 \begin{align*} \frac{\sqrt{2}{\left (B a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} + 6 \, A a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} + 3 \, B a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - \frac{{\left (6 \, A a^{3} e^{x} + 3 \, B a^{3} e^{x} + B a^{3}\right )} e^{\left (-\frac{3}{2} \, x\right )}}{a^{\frac{3}{2}}}\right )}}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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