Optimal. Leaf size=68 \[ -\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}-\frac{2 d E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
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Rubi [A] time = 0.0391602, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3768, 3771, 2639} \[ -\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}-\frac{2 d E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \csc (e+f x) \sqrt{d \csc (e+f x)} \, dx &=\frac{\int (d \csc (e+f x))^{3/2} \, dx}{d}\\ &=-\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}-d \int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx\\ &=-\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}-\frac{d \int \sqrt{\sin (e+f x)} \, dx}{\sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}-\frac{2 d E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right )}{f \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.128092, size = 57, normalized size = 0.84 \[ \frac{(d \csc (e+f x))^{3/2} \left (2 \sin ^{\frac{3}{2}}(e+f x) E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )-\sin (2 (e+f x))\right )}{d f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.121, size = 514, normalized size = 7.6 \begin{align*}{\frac{\sqrt{2}}{f} \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 ) \sqrt{{\frac{d}{\sin \left ( fx+e \right ) }}}} \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 \sqrt{d \csc \left (f x + e\right )} \csc \left (f x + e\right )\,{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 (\sqrt{d \csc \left (f x + e\right )} \csc \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \csc{\left (e + f x \right )}} \csc{\left (e + f x \right )}\, 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 \sqrt{d \csc \left (f x + e\right )} \csc \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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