Optimal. Leaf size=74 \[ -\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {2 \sqrt {d \csc (e+f x)} F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {\sin (e+f x)}}{3 f} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, 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,
2720} \begin {gather*} \frac {2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2720
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \csc ^2(e+f x) \sqrt {d \csc (e+f x)} \, dx &=\frac {\int (d \csc (e+f x))^{5/2} \, dx}{d^2}\\ &=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {1}{3} \int \sqrt {d \csc (e+f x)} \, dx\\ &=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {1}{3} \left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx\\ &=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}+\frac {2 \sqrt {d \csc (e+f x)} F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {\sin (e+f x)}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 55, normalized size = 0.74 \begin {gather*} -\frac {2 (d \csc (e+f x))^{3/2} \left (\cos (e+f x)+F\left (\left .\frac {1}{4} (-2 e+\pi -2 f x)\right |2\right ) \sin ^{\frac {3}{2}}(e+f x)\right )}{3 d f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.12, size = 319, normalized size = 4.31
method | result | size |
default | \(\frac {\sqrt {\frac {d}{\sin \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (i \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sin \left (f x +e \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+i \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sin \left (f x +e \right ) \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}-\cos \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{3 f \sin \left (f x +e \right )^{5}}\) | \(319\) |
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 = 0.10, size = 105, normalized size = 1.42 \begin {gather*} \frac {-i \, \sqrt {2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + i \, \sqrt {-2 i \, d} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, \sqrt {\frac {d}{\sin \left (f x + e\right )}} \cos \left (f x + e\right )}{3 \, f \sin \left (f x + e\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 \sqrt {d \csc {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, 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 {\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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