Optimal. Leaf size=75 \[ \frac {6 d E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 75, 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, 3769, 3771, 2639} \[ \frac {6 d E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \sqrt {d \csc (e+f x)} \sin ^3(e+f x) \, dx &=d^3 \int \frac {1}{(d \csc (e+f x))^{5/2}} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {1}{5} (3 d) \int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {(3 d) \int \sqrt {\sin (e+f x)} \, dx}{5 \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ &=-\frac {2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {6 d E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 62, normalized size = 0.83 \[ -\frac {2 \sqrt {d \csc (e+f x)} \left (\sin ^2(e+f x) \cos (e+f x)+3 \sqrt {\sin (e+f x)} E\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{5 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right ), x\right ) \]
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 \[ \int \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 538, normalized size = 7.17 \[ \frac {\sqrt {\frac {d}{\sin \left (f x +e \right )}}\, \left (-6 \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 )+\sin \left (f x +e \right )-i}{\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 )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+3 \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 )+\sin \left (f x +e \right )-i}{\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 )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}-6 \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 )+\sin \left (f x +e \right )-i}{\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 )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+3 \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 )+\sin \left (f x +e \right )-i}{\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 )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-4 \cos \left (f x +e \right ) \sqrt {2}+3 \sqrt {2}\right ) \sqrt {2}}{5 f} \]
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 \[ \int \sqrt {d \csc \left (f x + e\right )} \sin \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^3\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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