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 = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 2646
Rule 2751
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*}
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Mathematica [A] time = 0.04, 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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fricas [B] time = 0.49, size = 100, normalized size = 2.50 \[ \frac {\sqrt {\frac {1}{2}} {\left (B \cosh \relax (x)^{3} + B \sinh \relax (x)^{3} + 3 \, {\left (2 \, A + B\right )} \cosh \relax (x)^{2} + 3 \, {\left (B \cosh \relax (x) + 2 \, A + B\right )} \sinh \relax (x)^{2} - 3 \, {\left (2 \, A + B\right )} \cosh \relax (x) + 3 \, {\left (B \cosh \relax (x)^{2} + 2 \, {\left (2 \, A + B\right )} \cosh \relax (x) - 2 \, A - B\right )} \sinh \relax (x) - B\right )} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{3 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 71, normalized size = 1.78 \[ -\frac {1}{6} \, \sqrt {2} {\left (\frac {{\left (6 \, A a^{2} e^{x} + 3 \, B a^{2} e^{x} + B a^{2}\right )} e^{\left (-\frac {3}{2} \, x\right )}}{a^{\frac {3}{2}}} - \frac {B a^{\frac {7}{2}} e^{\left (\frac {3}{2} \, x\right )} + 6 \, A a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, x\right )} + 3 \, B a^{\frac {7}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 39, normalized size = 0.98 \[ \frac {2 \cosh \left (\frac {x}{2}\right ) a \sinh \left (\frac {x}{2}\right ) \left (2 B \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+3 A +B \right ) \sqrt {2}}{3 \sqrt {a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 90, normalized size = 2.25 \[ {\left (\sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, x\right )} - \sqrt {2} \sqrt {a} e^{\left (-\frac {1}{2} \, x\right )}\right )} A + \frac {1}{6} \, {\left ({\left (\sqrt {2} \sqrt {a} e^{\left (-x\right )} + 3 \, \sqrt {2} \sqrt {a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac {5}{2} \, x\right )} - {\left (3 \, \sqrt {2} \sqrt {a} e^{\left (-x\right )} + \sqrt {2} \sqrt {a} e^{\left (-2 \, x\right )}\right )} e^{\left (\frac {1}{2} \, x\right )}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \left (A+B\,\mathrm {cosh}\relax (x)\right )\,\sqrt {a+a\,\mathrm {cosh}\relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \left (\cosh {\relax (x )} + 1\right )} \left (A + B \cosh {\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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