Optimal. Leaf size=75 \[ \frac {2 d \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}} \]
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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, 2641} \[ \frac {2 d \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 f}-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx &=d^3 \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {1}{3} d \int \sqrt {d \csc (e+f x)} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {1}{3} \left (d \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx\\ &=-\frac {2 d^2 \cos (e+f x)}{3 f \sqrt {d \csc (e+f x)}}+\frac {2 d \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 = 0.06, size = 56, normalized size = 0.75 \[ -\frac {d \sqrt {d \csc (e+f x)} \left (\sin (2 (e+f x))+2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (d \cos \left (f x + e\right )^{2} - d\right )} \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right ) \sin \left (f x + e\right ), x\right ) \]
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 \[ \int \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 189, normalized size = 2.52 \[ -\frac {\left (i \sin \left (f x +e \right ) \sqrt {-\frac {i \cos \left (f x +e \right )-\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}+\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-\cos \left (f x +e \right ) \sqrt {2}\right ) \left (\frac {d}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {2}}{3 f \left (-1+\cos \left (f x +e \right )\right )} \]
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 \[ \int \left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}} \sin \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^3\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
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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