Optimal. Leaf size=71 \[ -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sin ^{\frac {3}{2}}(x) F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3768, 3771, 2641} \[ -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sin ^{\frac {3}{2}}(x) F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \left (a \csc ^3(x)\right )^{3/2} \, dx &=-\frac {\left (a \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{9/2} \, dx}{(-\csc (x))^{3/2}}\\ &=-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {\left (5 a \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{5/2} \, dx}{7 (-\csc (x))^{3/2}}\\ &=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {\left (5 a \sqrt {a \csc ^3(x)}\right ) \int \sqrt {-\csc (x)} \, dx}{21 (-\csc (x))^{3/2}}\\ &=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}+\frac {1}{21} \left (5 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sin (x)}} \, dx\\ &=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} F\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 = 0.11, size = 46, normalized size = 0.65 \[ -\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (22 \sin (2 x)-5 \sin (4 x)+40 \sin ^{\frac {9}{2}}(x) F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \csc \relax (x)^{3}} a \csc \relax (x)^{3}, x\right ) \]
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 \[ \int \left (a \csc \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 372, normalized size = 5.24 \[ -\frac {\left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (5 i \sin \relax (x ) \left (\cos ^{3}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}+5 i \sin \relax (x ) \left (\cos ^{2}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-5 i \sin \relax (x ) \cos \relax (x ) \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}-5 i \sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )-i+\sin \relax (x )}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sin \relax (x )-10 \left (\cos ^{3}\relax (x )\right )+16 \cos \relax (x )\right ) \left (-\frac {2 a}{\sin \relax (x ) \left (-1+\cos ^{2}\relax (x )\right )}\right )^{\frac {3}{2}} \sqrt {8}}{168 \sin \relax (x )^{3}} \]
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 \[ \int \left (a \csc \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (\frac {a}{{\sin \relax (x)}^3}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \csc ^{3}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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