Optimal. Leaf size=44 \[ \frac{2}{3} B \sinh (x) \sqrt{a-a \cosh (x)}-\frac{2 a (3 A-B) \sinh (x)}{3 \sqrt{a-a \cosh (x)}} \]
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Rubi [A] time = 0.0576204, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2751, 2646} \[ \frac{2}{3} B \sinh (x) \sqrt{a-a \cosh (x)}-\frac{2 a (3 A-B) \sinh (x)}{3 \sqrt{a-a \cosh (x)}} \]
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 = 0.0506901, size = 32, normalized size = 0.73 \[ \frac{2}{3} \coth \left (\frac{x}{2}\right ) \sqrt{a-a \cosh (x)} (3 A+B \cosh (x)-2 B) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 39, normalized size = 0.9 \begin{align*} -{\frac{4\,a}{3}\sinh \left ({\frac{x}{2}} \right ) \cosh \left ({\frac{x}{2}} \right ) \left ( 2\,B \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+3\,A-3\,B \right ){\frac{1}{\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6375, size = 147, normalized size = 3.34 \begin{align*} -{\left (\frac{\sqrt{2} \sqrt{a} e^{\left (-x\right )}}{\sqrt{-e^{\left (-x\right )}}} + \frac{\sqrt{2} \sqrt{a}}{\sqrt{-e^{\left (-x\right )}}}\right )} A + \frac{1}{6} \,{\left (\frac{{\left (3 \, \sqrt{2} \sqrt{a} e^{\left (-x\right )} - \sqrt{2} \sqrt{a}\right )} e^{x}}{\sqrt{-e^{\left (-x\right )}}} + \frac{3 \, \sqrt{2} \sqrt{a} e^{\left (-x\right )} - \sqrt{2} \sqrt{a} e^{\left (-2 \, x\right )}}{\sqrt{-e^{\left (-x\right )}}}\right )} B \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14688, size = 319, normalized size = 7.25 \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 = 1.23306, size = 166, normalized size = 3.77 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{-a e^{x}} B a e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 6 \, \sqrt{-a e^{x}} A a \mathrm{sgn}\left (-e^{x} + 1\right ) - 3 \, \sqrt{-a e^{x}} B a \mathrm{sgn}\left (-e^{x} + 1\right ) - \frac{{\left (6 \, A a^{3} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{3} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + B a^{3} \mathrm{sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-x\right )}}{\sqrt{-a e^{x}} a}\right )}}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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