Optimal. Leaf size=104 \[ \frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{2 a c f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.443844, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2841, 2743, 2741, 3770} \[ \frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 a^2 c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{2 a c f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2743
Rule 2741
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x)}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx &=\frac{\int \frac{1}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx}{a c}\\ &=-\frac{\cos (e+f x)}{2 a c f (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}+\frac{\int \frac{1}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx}{2 a^2 c}\\ &=-\frac{\cos (e+f x)}{2 a c f (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) \int \sec (e+f x) \, dx}{2 a^2 c \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{\cos (e+f x)}{2 a c f (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}+\frac{\tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{2 a^2 c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.775292, size = 163, normalized size = 1.57 \[ \frac{\cos ^3(e+f x) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+\sin (e+f x) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )+1\right )}{2 c f (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.187, size = 244, normalized size = 2.4 \begin{align*} -{\frac{ \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}-\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +\cos \left ( fx+e \right ) +2\,\sin \left ( fx+e \right ) -2 \right ) \left ( \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -2 \right ) }{4\,f \left ( -1+\cos \left ( fx+e \right ) \right ) } \left ( \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 ) -\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 ) -\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 ) -\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \right ) \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{5}{2}}} \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{\cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\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 [A] time = 2.01209, size = 814, normalized size = 7.83 \begin{align*} \left [\frac{\sqrt{a c}{\left (\cos \left (f x + e\right ) \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} \log \left (-\frac{a c \cos \left (f x + e\right )^{3} - 2 \, a c \cos \left (f x + e\right ) - 2 \, \sqrt{a c} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3}}\right ) - 2 \, \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{4 \,{\left (a^{3} c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )\right )}}, -\frac{\sqrt{-a c}{\left (\cos \left (f x + e\right ) \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} \arctan \left (\frac{\sqrt{-a c} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{2 \,{\left (a^{3} c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\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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