Optimal. Leaf size=140 \[ \frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.270781, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2743, 2741, 3770} \[ \frac{\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{4 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a f (a \sin (e+f x)+a)^{3/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2743
Rule 2741
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}} \, dx &=-\frac{\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}+\frac{\int \frac{1}{(a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}} \, dx}{2 a}\\ &=-\frac{\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a 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}{4 a^2}\\ &=-\frac{\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a 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}{4 a^2 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}-\frac{\cos (e+f x)}{4 a 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)}{4 a^2 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.628072, size = 211, normalized size = 1.51 \[ \frac{\cos (e+f x) \left (-3 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+\cos (2 (e+f x)) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )+3 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+\sin (e+f x) \left (-4 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+4 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )-2\right )-4\right )}{8 f (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.181, size = 252, normalized size = 1.8 \begin{align*} -{\frac{\cos \left ( fx+e \right ) }{4\,f} \left ( - \left ( \cos \left ( fx+e \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +2\,\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 ) -2\,\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 ) +2\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-3\,\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 ) -2\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -2 \right ) \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-c \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}} \sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32909, size = 990, normalized size = 7.07 \begin{align*} \left [\frac{{\left (\cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )\right )} \sqrt{a c} \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}{\left (\sin \left (f x + e\right ) + 2\right )}}{8 \,{\left (a^{3} c f \cos \left (f x + e\right )^{3} - 2 \, a^{3} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} c f \cos \left (f x + e\right )\right )}}, -\frac{{\left (\cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )\right )} \sqrt{-a c} \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}{\left (\sin \left (f x + e\right ) + 2\right )}}{4 \,{\left (a^{3} c f \cos \left (f x + e\right )^{3} - 2 \, a^{3} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} c 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{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}} \sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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