Optimal. Leaf size=181 \[ \frac {2 i \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{15} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2} \]
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Rubi [A] time = 0.32, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {2 i \left (a^2-b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {a+b \cosh (x)}}-\frac {2 i \left (3 a^2 B+20 a A b+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2}{15} \sinh (x) (3 a B+5 A b) \sqrt {a+b \cosh (x)}+\frac {2}{5} B \sinh (x) (a+b \cosh (x))^{3/2} \]
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 (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx &=\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)+\frac {2}{5} \int \sqrt {a+b \cosh (x)} \left (\frac {1}{2} (5 a A+3 b B)+\frac {1}{2} (5 A b+3 a B) \cosh (x)\right ) \, dx\\ &=\frac {2}{15} (5 A b+3 a B) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)+\frac {4}{15} \int \frac {\frac {1}{4} \left (15 a^2 A+5 A b^2+12 a b B\right )+\frac {1}{4} \left (20 a A b+3 a^2 B+9 b^2 B\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx\\ &=\frac {2}{15} (5 A b+3 a B) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)-\frac {\left (\left (a^2-b^2\right ) (5 A b+3 a B)\right ) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{15 b}+\frac {\left (20 a A b+3 a^2 B+9 b^2 B\right ) \int \sqrt {a+b \cosh (x)} \, dx}{15 b}\\ &=\frac {2}{15} (5 A b+3 a B) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)+\frac {\left (\left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{15 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) (5 A b+3 a 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}{15 b \sqrt {a+b \cosh (x)}}\\ &=-\frac {2 i \left (20 a A b+3 a^2 B+9 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \left (a^2-b^2\right ) (5 A b+3 a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{15 b \sqrt {a+b \cosh (x)}}+\frac {2}{15} (5 A b+3 a B) \sqrt {a+b \cosh (x)} \sinh (x)+\frac {2}{5} B (a+b \cosh (x))^{3/2} \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.70, size = 124, normalized size = 0.69 \[ \frac {2}{15} \sqrt {a+b \cosh (x)} \left (\sinh (x) (6 a B+5 A b+3 b B \cosh (x))-\frac {i \left (\left (3 a^2 B+20 a A b+9 b^2 B\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-(a-b) (3 a B+5 A b) F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cosh \relax (x)^{2} + A a + {\left (B a + A b\right )} \cosh \relax (x)\right )} \sqrt {b \cosh \relax (x) + a}, x\right ) \]
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 \[ \int {\left (B \cosh \relax (x) + A\right )} {\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 973, normalized size = 5.38 \[ \text {result too large to display} \]
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 \[ \int {\left (B \cosh \relax (x) + A\right )} {\left (b \cosh \relax (x) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \left (A+B\,\mathrm {cosh}\relax (x)\right )\,{\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]
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 \[ \int \left (A + B \cosh {\relax (x )}\right ) \left (a + b \cosh {\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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