Optimal. Leaf size=251 \[ -\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 i (A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}} \]
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Rubi [A] time = 0.33, antiderivative size = 251, 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} \[ -\frac {2 \cosh (x) \left (a^2 (-B)+4 a A b+3 b^2 B\right )}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}-\frac {2 \cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 i (A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}+\frac {2 i \left (a^2 (-B)+4 a A b+3 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}} \]
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 \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx &=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} (a A+b B)+\frac {1}{2} (A b-a B) \sinh (x)}{(a+b \sinh (x))^{3/2}} \, dx}{3 \left (a^2+b^2\right )}\\ &=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (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 ) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx}{3 \left (a^2+b^2\right )^2}\\ &=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}-\frac {(A b-a B) \int \frac {1}{\sqrt {a+b \sinh (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 \sinh (x)} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {\left (\left (4 a A b-a^2 B+3 b^2 B\right ) \sqrt {a+b \sinh (x)}\right ) \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left ((A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}\\ &=-\frac {2 (A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^{3/2}}-\frac {2 \left (4 a A b-a^2 B+3 b^2 B\right ) \cosh (x)}{3 \left (a^2+b^2\right )^2 \sqrt {a+b \sinh (x)}}+\frac {2 i \left (4 a A b-a^2 B+3 b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 b \left (a^2+b^2\right )^2 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i (A b-a B) F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 b \left (a^2+b^2\right ) \sqrt {a+b \sinh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 236, normalized size = 0.94 \[ \frac {2 i \left (i b \cosh (x) \left (-\left (a^2 B-4 a A b-3 b^2 B\right ) (a+b \sinh (x))-\left (\left (a^2+b^2\right ) (a B-A b)\right )\right )+\sqrt {\frac {a+b \sinh (x)}{a-i b}} (a+b \sinh (x)) \left (b \left (3 a^2 A+4 a b B-A b^2\right ) F\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )+\left (a^2 (-B)+4 a A b+3 b^2 B\right ) \left ((a-i b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-a F\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )\right )\right )\right )}{3 b \left (a^2+b^2\right )^2 (a+b \sinh (x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \sinh \relax (x) + A\right )} \sqrt {b \sinh \relax (x) + a}}{b^{3} \sinh \relax (x)^{3} + 3 \, a b^{2} \sinh \relax (x)^{2} + 3 \, a^{2} b \sinh \relax (x) + a^{3}}, x\right ) \]
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 \[ \int \frac {B \sinh \relax (x) + A}{{\left (b \sinh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 806, normalized size = 3.21 \[ \frac {\sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}\, \left (\frac {B \left (-\frac {2 b \left (\cosh ^{2}\relax (x )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {2 a \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {2 b \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \left (\left (-\frac {a}{b}-i\right ) \EllipticE \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{\left (a^{2}+b^{2}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}\right )}{b}+\frac {\left (A b -a B \right ) \left (-\frac {2 \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}{3 b \left (a^{2}+b^{2}\right ) \left (\sinh \relax (x )+\frac {a}{b}\right )^{2}}-\frac {8 b \left (\cosh ^{2}\relax (x )\right ) a}{3 \left (a^{2}+b^{2}\right )^{2} \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {2 \left (3 a^{2}-b^{2}\right ) \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )}{\left (3 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}+\frac {8 a b \left (\frac {a}{b}-i\right ) \sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \left (\left (-\frac {a}{b}-i\right ) \EllipticE \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )+i \EllipticF \left (\sqrt {\frac {-b \sinh \relax (x )-a}{i b -a}}, \sqrt {\frac {-i b +a}{i b +a}}\right )\right )}{3 \left (a^{2}+b^{2}\right )^{2} \sqrt {\left (a +b \sinh \relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}}\right )}{b}\right )}{\cosh \relax (x ) \sqrt {a +b \sinh \relax (x )}} \]
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 \[ \int \frac {B \sinh \relax (x) + A}{{\left (b \sinh \relax (x) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,\mathrm {sinh}\relax (x)}{{\left (a+b\,\mathrm {sinh}\relax (x)\right )}^{5/2}} \,d x \]
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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