Optimal. Leaf size=123 \[ -\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}} \]
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Rubi [A]
time = 0.04, 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 = {4208, 3854,
3856, 2720} \begin {gather*} -\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \sin ^5(x) \cos (x)}{15 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \sin ^3(x) \cos (x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {78 \sin (x) \cos (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{77 a^2 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3854
Rule 3856
Rule 4208
Rubi steps
\begin {align*} \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx &=\frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{15/2}} \, dx}{a^2 \sqrt {a \csc ^3(x)}}\\ &=-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}}+\frac {\left (13 (-\csc (x))^{3/2}\right ) \int \frac {1}{(-\csc (x))^{11/2}} \, dx}{15 a^2 \sqrt {a \csc ^3(x)}}\\ &=-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}}+\frac {\left (39 (-\csc (x))^{3/2}\right ) \int \frac {1}{(-\csc (x))^{7/2}} \, dx}{55 a^2 \sqrt {a \csc ^3(x)}}\\ &=-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}}+\frac {\left (39 (-\csc (x))^{3/2}\right ) \int \frac {1}{(-\csc (x))^{3/2}} \, dx}{77 a^2 \sqrt {a \csc ^3(x)}}\\ &=-\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}}+\frac {\left (13 (-\csc (x))^{3/2}\right ) \int \sqrt {-\csc (x)} \, dx}{77 a^2 \sqrt {a \csc ^3(x)}}\\ &=-\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}}+\frac {13 \int \frac {1}{\sqrt {\sin (x)}} \, dx}{77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {26 \cot (x)}{77 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \csc ^3(x)}}-\frac {26 \cos (x) \sin ^3(x)}{165 a^2 \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^5(x)}{15 a^2 \sqrt {a \csc ^3(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 63, normalized size = 0.51 \begin {gather*} -\frac {\sqrt {a \csc ^3(x)} \sin (x) \left (24960 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sqrt {\sin (x)}+19122 \sin (2 x)-4406 \sin (4 x)+826 \sin (6 x)-77 \sin (8 x)\right )}{73920 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.18, size = 158, normalized size = 1.28
method | result | size |
default | \(-\frac {2 \left (-154 \left (\cos ^{8}\left (x \right )\right )+195 i \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {2}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (\cos \left (x \right )-1\right )}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )+154 \left (\cos ^{7}\left (x \right )\right )+644 \left (\cos ^{6}\left (x \right )\right )-644 \left (\cos ^{5}\left (x \right )\right )-1060 \left (\cos ^{4}\left (x \right )\right )+1060 \left (\cos ^{3}\left (x \right )\right )+960 \left (\cos ^{2}\left (x \right )\right )-960 \cos \left (x \right )\right ) \sqrt {8}}{1155 \left (\cos \left (x \right )-1\right ) \left (-\frac {2 a}{\sin \left (x \right ) \left (\cos ^{2}\left (x \right )-1\right )}\right )^{\frac {5}{2}} \sin \left (x \right )^{7}}\) | \(158\) |
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 = 1.19, size = 88, normalized size = 0.72 \begin {gather*} -\frac {2 \, {\left (77 \, \cos \left (x\right )^{9} - 399 \, \cos \left (x\right )^{7} + 852 \, \cos \left (x\right )^{5} - 1010 \, \cos \left (x\right )^{3} + 480 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} + 195 i \, \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{1155 \, a^{3}} \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 \frac {1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac {5}{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 \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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