Optimal. Leaf size=138 \[ \frac {2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {a+b \cosh (x)}}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)} \]
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Rubi [A] time = 0.21, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {a+b \cosh (x)}}-\frac {2 i (a B+3 A b) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{3} B \sinh (x) \sqrt {a+b \cosh (x)} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rubi steps
\begin {align*} \int \sqrt {a+b \cosh (x)} (A+B \cosh (x)) \, dx &=\frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{3} \int \frac {\frac {1}{2} (3 a A+b B)+\frac {1}{2} (3 A b+a B) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx\\ &=\frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)-\frac {\left (\left (a^2-b^2\right ) B\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{3 b}+\frac {(3 A b+a B) \int \sqrt {a+b \cosh (x)} \, dx}{3 b}\\ &=\frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)+\frac {\left ((3 A b+a B) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{3 b \sqrt {a+b \cosh (x)}}\\ &=-\frac {2 i (3 A b+a B) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) B \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \sqrt {a+b \cosh (x)}}+\frac {2}{3} B \sqrt {a+b \cosh (x)} \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.35, size = 123, normalized size = 0.89 \[ \frac {2 i B \left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-2 i (a+b) (a B+3 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+2 b B \sinh (x) (a+b \cosh (x))}{3 b \sqrt {a+b \cosh (x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \cosh \relax (x) + A\right )} \sqrt {b \cosh \relax (x) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \cosh \relax (x) + A\right )} \sqrt {b \cosh \relax (x) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 605, normalized size = 4.38 \[ \frac {2 \left (4 B \sqrt {-\frac {2 b}{a -b}}\, b \cosh \left (\frac {x}{2}\right ) \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (2 B \sqrt {-\frac {2 b}{a -b}}\, a +2 B \sqrt {-\frac {2 b}{a -b}}\, b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+3 A a \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+3 A b \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-6 A \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) b +a B \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+b B \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-2 B \sqrt {\frac {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}{a -b}+\frac {a +b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a \right ) \sqrt {\left (2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{3 \sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 b \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (a +b \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \cosh \relax (x) + A\right )} \sqrt {b \cosh \relax (x) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\mathrm {cosh}\relax (x)\right )\,\sqrt {a+b\,\mathrm {cosh}\relax (x)} \,d x \]
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 \[ \int \left (A + B \cosh {\relax (x )}\right ) \sqrt {a + b \cosh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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