Optimal. Leaf size=167 \[ \frac{(A+C) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.651349, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {3036, 2969, 2737, 2667, 31} \[ \frac{(A+C) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3036
Rule 2969
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{A+C \sin ^2(e+f x)}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx &=\frac{(A+C) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{\int \frac{-2 a^2 (A-C)+4 a^2 C \sin (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx}{4 a^2 c}\\ &=\frac{(A+C) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac{(A-3 C) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx}{4 a c}+\frac{(A+C) \int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx}{4 c^2}\\ &=\frac{(A+C) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac{((A-3 C) \cos (e+f x)) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{4 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{(a (A+C) \cos (e+f x)) \int \frac{\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{4 c \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+C) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{((A-3 C) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{4 c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{((A+C) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (e+f x)\right )}{4 c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+C) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{(A+C) \cos (e+f x) \log (1+\sin (e+f x))}{4 c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.628868, size = 190, normalized size = 1.14 \[ \frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (-(A-3 C) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+(A+C) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+A+C\right )}{2 f \sqrt{a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.392, size = 341, normalized size = 2. \begin{align*}{\frac{\cos \left ( fx+e \right ) }{2\,f} \left ( A\sin \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -A\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \sin \left ( fx+e \right ) +2\,C\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) \sin \left ( fx+e \right ) -3\,C\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \sin \left ( fx+e \right ) -C\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \sin \left ( fx+e \right ) +A\sin \left ( fx+e \right ) -A\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +A\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -2\,C\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) +C\sin \left ( fx+e \right ) +3\,C\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +C\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \right ){\frac{1}{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \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{C \sin \left (f x + e\right )^{2} + A}{\sqrt{a \sin \left (f x + e\right ) + a}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{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{{\left (C \cos \left (f x + e\right )^{2} - A - C\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{a c^{2} \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - a c^{2} \cos \left (f x + e\right )^{2}}, 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{A + C \sin ^{2}{\left (e + f x \right )}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \left (- c \left (\sin{\left (e + f x \right )} - 1\right )\right )^{\frac{3}{2}}}\, 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{C \sin \left (f x + e\right )^{2} + A}{\sqrt{a \sin \left (f x + e\right ) + a}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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