Optimal. Leaf size=73 \[ -\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{d^2 f}-\frac{2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
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Rubi [A] time = 0.0388136, antiderivative size = 73, 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, 3768, 3771, 2639} \[ -\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{d^2 f}-\frac{2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2639
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*}
Mathematica [A] time = 0.0513571, size = 55, normalized size = 0.75 \[ \frac{\frac{2 E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )}{\sqrt{\sin (e+f x)}}-2 \cot (e+f x)}{d f \sqrt{d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.115, size = 522, normalized size = 7.2 \begin{align*}{\frac{\sqrt{2}}{f \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \left ( 2\,\cos \left ( fx+e \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \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 EllipticE} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) -\cos \left ( fx+e \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \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 ) +2\,\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \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 EllipticE} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},1/2\,\sqrt{2} \right ) -\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \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} \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 \frac{\csc \left (f x + e\right )^{3}}{\left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}}}\,{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 (\frac{\sqrt{d \csc \left (f x + e\right )} \csc \left (f x + e\right )}{d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (e + f x \right )}}{\left (d \csc{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \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 \frac{\csc \left (f x + e\right )^{3}}{\left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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