Optimal. Leaf size=71 \[ -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \]
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Rubi [A]
time = 0.03, 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 = {4208, 3853,
3856, 2720} \begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3853
Rule 3856
Rule 4208
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 = 0.13, size = 46, normalized size = 0.65 \begin {gather*} -\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (40 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {9}{2}}(x)+22 \sin (2 x)-5 \sin (4 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.19, size = 372, normalized size = 5.24
method | result | size |
default | \(-\frac {\left (\cos \left (x \right )+1\right )^{2} \left (\cos \left (x \right )-1\right )^{2} \left (5 i \sin \left (x \right ) \left (\cos ^{3}\left (x \right )\right ) \sqrt {2}\, \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )+5 i \sin \left (x \right ) \left (\cos ^{2}\left (x \right )\right ) \sqrt {2}\, \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )-5 i \sin \left (x \right ) \cos \left (x \right ) \sqrt {2}\, \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )-5 i \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )-10 \left (\cos ^{3}\left (x \right )\right )+16 \cos \left (x \right )\right ) \left (-\frac {2 a}{\sin \left (x \right ) \left (\cos ^{2}\left (x \right )-1\right )}\right )^{\frac {3}{2}} \sqrt {8}}{168 \sin \left (x \right )^{3}}\) | \(372\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.10, size = 99, normalized size = 1.39 \begin {gather*} -\frac {5 \, {\left (i \, a \cos \left (x\right )^{2} - i \, a\right )} \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (-i \, a \cos \left (x\right )^{2} + i \, a\right )} \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{3} - 8 \, a \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{21 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \end {gather*}
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 \begin {gather*} \int \left (a \csc ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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