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 = 0.0531429, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, 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 3769
Rule 3771
Rule 2641
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*}
Mathematica [A] time = 0.0687143, 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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Maple [C] time = 0.118, size = 189, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3\,f \left ( -1+\cos \left ( fx+e \right ) \right ) } \left ( i\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}-\sqrt{2}\cos \left ( fx+e \right ) \right ) \left ({\frac{d}{\sin \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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