Optimal. Leaf size=100 \[ \frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{a f}-\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.447335, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2942, 2737, 2667, 31, 2948, 3475} \[ \frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)} \log (\sin (e+f x))}{a f}-\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2942
Rule 2737
Rule 2667
Rule 31
Rule 2948
Rule 3475
Rubi steps
\begin{align*} \int \frac{\csc (e+f x) \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx &=\frac{\int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)} \, dx}{a}-\int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=-\frac{(a c \cos (e+f x)) \int \frac{\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{\left (\sec (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}\right ) \int \cot (e+f x) \, dx}{a}\\ &=\frac{\log (\sin (e+f x)) \sec (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}{a f}-\frac{(c \cos (e+f x)) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (e+f x)\right )}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{c \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{\log (\sin (e+f x)) \sec (e+f x) \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}{a f}\\ \end{align*}
Mathematica [C] time = 1.31986, size = 145, normalized size = 1.45 \[ \frac{\sqrt{2} \left (e^{i (e+f x)}+i\right ) \sqrt{c-c \sin (e+f x)} \left (2 \tan ^{-1}\left (e^{i (e+f x)}\right )-i \left (\log \left (1-e^{2 i (e+f x)}\right )-\log \left (1+e^{2 i (e+f x)}\right )\right )\right )}{f \left (e^{i (e+f x)}-i\right ) \sqrt{-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.277, size = 113, normalized size = 1.1 \begin{align*}{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{f \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( 2\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \right ){\frac{1}{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53199, size = 80, normalized size = 0.8 \begin{align*} \frac{\frac{2 \, \sqrt{c} \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{\sqrt{a}} - \frac{\sqrt{c} \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{\sqrt{a}}}{f} \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{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{a \cos \left (f x + e\right )^{2} - a \sin \left (f x + e\right ) - a}, 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{\sqrt{- c \left (\sin{\left (e + f x \right )} - 1\right )}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \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{\sqrt{-c \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a} \sin \left (f x + e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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