Optimal. Leaf size=222 \[ -\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a f \sqrt{a+b \sin (e+f x)}} \]
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Rubi [A] time = 0.495332, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2802, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a f \sqrt{a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2802
Rule 3060
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{\int \frac{\csc (e+f x) \left (-\frac{b}{2}-\frac{1}{2} b \sin ^2(e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{a}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}-\frac{\int \sqrt{a+b \sin (e+f x)} \, dx}{2 a}-\frac{\int \frac{\csc (e+f x) \left (\frac{b^2}{2}-\frac{1}{2} a b \sin (e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{a b}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}+\frac{1}{2} \int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx-\frac{b \int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx}{2 a}-\frac{\sqrt{a+b \sin (e+f x)} \int \sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}} \, dx}{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}-\frac{E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (e+f x)}}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{\sqrt{\frac{a+b \sin (e+f x)}{a+b}} \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{2 \sqrt{a+b \sin (e+f x)}}-\frac{\left (b \sqrt{\frac{a+b \sin (e+f x)}{a+b}}\right ) \int \frac{\csc (e+f x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{2 a \sqrt{a+b \sin (e+f x)}}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{a f}-\frac{E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (e+f x)}}{a f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{F\left (\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)}}-\frac{b \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}}}{a f \sqrt{a+b \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 10.0247, size = 315, normalized size = 1.42 \[ \frac{-4 \cot (e+f x) \sqrt{a+b \sin (e+f x)}+\frac{6 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (e+f x)}}+\frac{2 i \sec (e+f x) \sqrt{-\frac{b (\sin (e+f x)-1)}{a+b}} \sqrt{-\frac{b (\sin (e+f x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )\right )}{a b \sqrt{-\frac{1}{a+b}}}}{4 a f} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.823, size = 412, normalized size = 1.9 \begin{align*}{\frac{1}{f\cos \left ( fx+e \right ) }\sqrt{- \left ( -b\sin \left ( fx+e \right ) -a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( -{\frac{1}{a\sin \left ( fx+e \right ) }\sqrt{- \left ( -b\sin \left ( fx+e \right ) -a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}-{\frac{b}{a} \left ({\frac{a}{b}}-1 \right ) \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}}\sqrt{{\frac{b \left ( 1-\sin \left ( fx+e \right ) \right ) }{a+b}}}\sqrt{{\frac{ \left ( -\sin \left ( fx+e \right ) -1 \right ) b}{a-b}}} \left ( \left ( -{\frac{a}{b}}-1 \right ){\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) +{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) \right ){\frac{1}{\sqrt{- \left ( -b\sin \left ( fx+e \right ) -a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}}+{\frac{{b}^{2}}{{a}^{2}} \left ({\frac{a}{b}}-1 \right ) \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}}\sqrt{{\frac{b \left ( 1-\sin \left ( fx+e \right ) \right ) }{a+b}}}\sqrt{{\frac{ \left ( -\sin \left ( fx+e \right ) -1 \right ) b}{a-b}}}{\it EllipticPi} \left ( \sqrt{{\frac{a+b\sin \left ( fx+e \right ) }{a-b}}},-{\frac{b}{a} \left ( -{\frac{a}{b}}+1 \right ) },\sqrt{{\frac{a-b}{a+b}}} \right ){\frac{1}{\sqrt{- \left ( -b\sin \left ( fx+e \right ) -a \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \right ){\frac{1}{\sqrt{a+b\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{\csc \left (f x + e\right )^{2}}{\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(-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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\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 )^{2}}{\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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