Optimal. Leaf size=231 \[ -\frac {2 \sinh (x) \left (a^2 (-B)+4 a A b-3 b^2 B\right )}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 \sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \left (a^2 (-B)+4 a A b-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]
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Rubi [A] time = 0.35, antiderivative size = 231, 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 = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 \sinh (x) \left (a^2 (-B)+4 a A b-3 b^2 B\right )}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 \sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \left (a^2 (-B)+4 a A b-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^{5/2}} \, dx &=-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} (a A-b B)+\frac {1}{2} (A b-a B) \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2 A+A b^2-4 a b B\right )+\frac {1}{4} \left (4 a A b-a^2 B-3 b^2 B\right ) \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {(A b-a B) \int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{3 b \left (a^2-b^2\right )}+\frac {\left (4 a A b-a^2 B-3 b^2 B\right ) \int \sqrt {a+b \cosh (x)} \, dx}{3 b \left (a^2-b^2\right )^2}\\ &=-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {\left (\left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\left ((A b-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}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}\\ &=-\frac {2 i \left (4 a A b-a^2 B-3 b^2 B\right ) \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i (A b-a B) \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 b \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 (A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B-3 b^2 B\right ) \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 172, normalized size = 0.74 \[ \frac {2 \left (\frac {\sinh (x) \left (2 a^3 B+b \cosh (x) \left (a^2 B-4 a A b+3 b^2 B\right )-5 a^2 A b+2 a b^2 B+A b^3\right )}{\left (a^2-b^2\right )^2}+\frac {i \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} \left (\left (a^2 B-4 a A b+3 b^2 B\right ) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-(a-b) (a B-A b) F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )\right )}{b (a-b)^2}\right )}{3 (a+b \cosh (x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cosh \relax (x) + A\right )} \sqrt {b \cosh \relax (x) + a}}{b^{3} \cosh \relax (x)^{3} + 3 \, a b^{2} \cosh \relax (x)^{2} + 3 \, a^{2} b \cosh \relax (x) + a^{3}}, x\right ) \]
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 \[ \int \frac {B \cosh \relax (x) + A}{{\left (b \cosh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.55, size = 797, normalized size = 3.45 \[ \text {result too large to display} \]
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 \[ \int \frac {B \cosh \relax (x) + A}{{\left (b \cosh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,\mathrm {cosh}\relax (x)}{{\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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