Optimal. Leaf size=43 \[ \frac{2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
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Rubi [A] time = 0.0185217, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3771, 2639} \[ \frac{2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx &=\frac{\int \sqrt{\sin (e+f x)} \, dx}{\sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ &=\frac{2 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 = 0.0152657, size = 42, normalized size = 0.98 \[ -\frac{2 E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.124, size = 521, normalized size = 12.1 \begin{align*} -{\frac{\sqrt{2}}{f\sin \left ( fx+e \right ) } \left ( 2\,\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\cos \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\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}-\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\cos \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 ) }}}{\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\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \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 ) }}}{\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{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}+\sqrt{2}\cos \left ( fx+e \right ) -\sqrt{2} \right ){\frac{1}{\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 \frac{1}{\sqrt{d \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 (\frac{\sqrt{d \csc \left (f x + e\right )}}{d \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 \frac{1}{\sqrt{d \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 \frac{1}{\sqrt{d \csc \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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