Optimal. Leaf size=177 \[ -\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]
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Rubi [A] time = 0.21, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {2664, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}+\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2664
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cosh (x))^{5/2}} \, dx &=-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3 a}{2}+\frac {1}{2} b \cosh (x)}{(a+b \cosh (x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {8 a b \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+b^2\right )+a b \cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {(4 a) \int \sqrt {a+b \cosh (x)} \, dx}{3 \left (a^2-b^2\right )^2}-\frac {\int \frac {1}{\sqrt {a+b \cosh (x)}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}+\frac {\left (4 a \sqrt {a+b \cosh (x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}} \, dx}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {\sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}} \, dx}{3 \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}\\ &=-\frac {8 i a \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 \left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 b \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^{3/2}}-\frac {8 a b \sinh (x)}{3 \left (a^2-b^2\right )^2 \sqrt {a+b \cosh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 135, normalized size = 0.76 \[ \frac {2 b \sinh (x) \left (-5 a^2-4 a b \cosh (x)+b^2\right )+2 i (a-b) (a+b)^2 \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} F\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-8 i a (a+b)^2 \left (\frac {a+b \cosh (x)}{a+b}\right )^{3/2} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{3 (a-b)^2 (a+b)^2 (a+b \cosh (x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cosh \relax (x) + a}}{b^{3} \cosh \relax (x)^{3} + 3 \, a b^{2} \cosh \relax (x)^{2} + 3 \, a^{2} b \cosh \relax (x) + a^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \cosh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.84, size = 459, normalized size = 2.59 \[ \frac {\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 )}\, \left (-\frac {\cosh \left (\frac {x}{2}\right ) \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 )}}{3 b \left (a -b \right ) \left (a +b \right ) \left (\cosh ^{2}\left (\frac {x}{2}\right )+\frac {a -b}{2 b}\right )^{2}}-\frac {16 b \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right ) a}{3 \left (a -b \right )^{2} \left (a +b \right )^{2} \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 )}}+\frac {2 \left (3 a -b \right ) \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+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 a +2 b}{b}}}{2}\right )}{\left (3 a^{3}+3 a^{2} b -3 a \,b^{2}-3 b^{3}\right ) \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 )}}-\frac {32 a b \left (-a +b \right ) \sqrt {\frac {2 b \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+a -b}{a -b}}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )-\EllipticE \left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {\frac {-2 a +2 b}{b}}}{2}\right )\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \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 )}\, \left (2 a -2 b \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 \[ \int \frac {1}{{\left (b \cosh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,\mathrm {cosh}\relax (x)\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \cosh {\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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