Optimal. Leaf size=73 \[ -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3853, 3856,
2719} \begin {gather*} -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{d f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2719
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx &=\frac {\int (d \csc (e+f x))^{3/2} \, dx}{d^3}\\ &=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {\int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx}{d}\\ &=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {\int \sqrt {\sin (e+f x)} \, dx}{d \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ &=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d^2 f}-\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{d f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.75 \begin {gather*} \frac {-2 \cot (e+f x)+\frac {2 E\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right )}{\sqrt {\sin (e+f x)}}}{d f \sqrt {d \csc (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.11, size = 510, normalized size = 6.99
method | result | size |
default | \(\frac {\left (2 \cos \left (f x +e \right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-i \cos \left (f x +e \right )+\sin \left (f x +e \right )+i}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-\cos \left (f x +e \right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-i \cos \left (f x +e \right )+\sin \left (f x +e \right )+i}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-i \cos \left (f x +e \right )+\sin \left (f x +e \right )+i}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-i \cos \left (f x +e \right )+\sin \left (f x +e \right )+i}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right ) \sqrt {2}}{f \left (\frac {d}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{2}}\) | \(510\) |
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 = 0.12, size = 89, normalized size = 1.22 \begin {gather*} -\frac {2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{d^{2} f} \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 {\csc ^{3}{\left (e + f x \right )}}{\left (d \csc {\left (e + f 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 \frac {1}{{\sin \left (e+f\,x\right )}^3\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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