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 = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 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 = 0.06, 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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fricas [B] time = 0.49, size = 107, normalized size = 2.43 \[ \frac {\sqrt {\frac {1}{2}} {\left (B \cosh \relax (x)^{3} + B \sinh \relax (x)^{3} + 3 \, {\left (2 \, A - B\right )} \cosh \relax (x)^{2} + 3 \, {\left (B \cosh \relax (x) + 2 \, A - B\right )} \sinh \relax (x)^{2} + 3 \, {\left (2 \, A - B\right )} \cosh \relax (x) + 3 \, {\left (B \cosh \relax (x)^{2} + 2 \, {\left (2 \, A - B\right )} \cosh \relax (x) + 2 \, A - B\right )} \sinh \relax (x) + B\right )} \sqrt {-\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{3 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 131, normalized size = 2.98 \[ \frac {1}{6} \, \sqrt {2} {\left (\frac {{\left (6 \, A a^{2} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{2} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + B a^{2} \mathrm {sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-x\right )}}{\sqrt {-a e^{x}} a} - \frac {\sqrt {-a e^{x}} B a^{3} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 6 \, \sqrt {-a e^{x}} A a^{3} \mathrm {sgn}\left (-e^{x} + 1\right ) - 3 \, \sqrt {-a e^{x}} B a^{3} \mathrm {sgn}\left (-e^{x} + 1\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 39, normalized size = 0.89 \[ -\frac {4 \sinh \left (\frac {x}{2}\right ) a \cosh \left (\frac {x}{2}\right ) \left (2 B \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+3 A -3 B \right )}{3 \sqrt {-2 a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 109, normalized size = 2.48 \[ -{\left (\frac {\sqrt {2} \sqrt {a} e^{\left (-x\right )}}{\sqrt {-e^{\left (-x\right )}}} + \frac {\sqrt {2} \sqrt {a}}{\sqrt {-e^{\left (-x\right )}}}\right )} A + \frac {1}{6} \, {\left (\frac {{\left (3 \, \sqrt {2} \sqrt {a} e^{\left (-x\right )} - \sqrt {2} \sqrt {a}\right )} e^{x}}{\sqrt {-e^{\left (-x\right )}}} + \frac {3 \, \sqrt {2} \sqrt {a} e^{\left (-x\right )} - \sqrt {2} \sqrt {a} e^{\left (-2 \, x\right )}}{\sqrt {-e^{\left (-x\right )}}}\right )} B \]
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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