Optimal. Leaf size=63 \[ \frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]
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Rubi [A] time = 0.129918, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2807, 2805} \[ \frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx &=\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} \int \frac{\csc (e+f x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{\sqrt{a+b \sin (e+f x)}}\\ &=\frac{2 \Pi \left (2;\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}{f \sqrt{a+b \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.0771257, size = 62, normalized size = 0.98 \[ -\frac{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.589, size = 135, normalized size = 2.1 \begin{align*} -2\,{\frac{a-b}{\cos \left ( fx+e \right ) a\sqrt{a+b\sin \left ( fx+e \right ) }f}\sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( fx+e \right ) \right ) b}{a-b}}}{\it EllipticPi} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},{\frac{a-b}{a}},\sqrt{{\frac{a-b}{a+b}}} \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{\csc \left (f x + e\right )}{\sqrt{b \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\left (e + f x \right )}}{\sqrt{a + b \sin{\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{\csc \left (f x + e\right )}{\sqrt{b \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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