Optimal. Leaf size=97 \[ -\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}} \]
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Rubi [A] time = 0.423393, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {2841, 2739, 2737, 2667, 31} \[ -\frac{c \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2739
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) \sqrt{c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx &=\frac{\int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{a c}\\ &=-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{\int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{(c \cos (e+f x)) \int \frac{\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{a \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{(c \cos (e+f x)) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \sin (e+f x)\right )}{a^2 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))}{a^2 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a+a \sin (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.887886, size = 100, normalized size = 1.03 \[ \frac{\sec (e+f x) \sqrt{c-c \sin (e+f x)} \left (-2 \log \left (e^{i (e+f x)}+i\right )+\left (i f x-2 \log \left (e^{i (e+f x)}+i\right )\right ) \sin (e+f x)+i f x-2\right )}{a^2 f \sqrt{a (\sin (e+f x)+1)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.213, size = 190, normalized size = 2. \begin{align*}{\frac{\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}{f \left ( -1+\cos \left ( fx+e \right ) +\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 ) \sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) \ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) -2\,\sin \left ( fx+e \right ) +2\,\ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) \right ) \sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{5}{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{\sqrt{-c \sin \left (f x + e\right ) + c} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{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{\sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c} \cos \left (f x + e\right )^{2}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\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{\sqrt{-c \sin \left (f x + e\right ) + c} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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