Optimal. Leaf size=123 \[ -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)}+\frac {154}{195} a^2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)} \]
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Rubi [A] time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3768, 3771, 2639} \[ -\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)}+\frac {154}{195} a^2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3768
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \left (a \csc ^3(x)\right )^{5/2} \, dx &=\frac {\left (a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{15/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (11 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{11/2} \, dx}{13 (-\csc (x))^{3/2}}\\ &=-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{7/2} \, dx}{117 (-\csc (x))^{3/2}}\\ &=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{3/2} \, dx}{195 (-\csc (x))^{3/2}}\\ &=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{195 (-\csc (x))^{3/2}}\\ &=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {1}{195} \left (77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx\\ &=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+\frac {154}{195} a^2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)\\ \end {align*}
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Mathematica [A] time = 0.18, size = 58, normalized size = 0.47 \[ \frac {\left (a \csc ^3(x)\right )^{5/2} \left (-9414 \sin (2 x)+5346 \sin (4 x)-1694 \sin (6 x)+231 \sin (8 x)+29568 \sin ^{\frac {15}{2}}(x) E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right )}{37440} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \csc \relax (x)^{3}} a^{2} \csc \relax (x)^{6}, x\right ) \]
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 \[ \int \left (a \csc \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.77, size = 1313, normalized size = 10.67 \[ \text {result too large to display} \]
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 \[ \int \left (a \csc \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (\frac {a}{{\sin \relax (x)}^3}\right )}^{5/2} \,d x \]
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 \[ \int \left (a \csc ^{3}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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