Optimal. Leaf size=193 \[ \frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.662676, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}} \]
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) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx &=\frac{\int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac{\int \frac{(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a \int \frac{(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c^3}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac{a^2 \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c^4}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac{\left (a^3 \cos (e+f x)\right ) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^3 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{\left (a^3 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^4 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac{a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.21542, size = 234, normalized size = 1.21 \[ \frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (3 \sin (3 (e+f x)) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+30 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-18 \cos (2 (e+f x)) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+1\right )-9 \sin (e+f x) \left (5 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+4\right )+34\right )}{6 c^4 f (\sin (e+f x)-1)^4 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.208, size = 401, normalized size = 2.1 \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}{3\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) - \left ( \cos \left ( fx+e \right ) \right ) ^{3}+2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4\,\sin \left ( fx+e \right ) +2\,\cos \left ( fx+e \right ) -4 \right ) } \left ( 6\,\sin \left ( fx+e \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 ) -3\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -18\, \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 ) +9\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) -24\,\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 ) +12\,\sin \left ( fx+e \right ) \ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) +6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+14\,\sin \left ( fx+e \right ) +24\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -12\,\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) -6 \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{9}{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{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{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 (a^{2} \cos \left (f x + e\right )^{4} - 2 \, a^{2} \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - 2 \, a^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{5 \, c^{5} \cos \left (f x + e\right )^{4} - 20 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5} -{\left (c^{5} \cos \left (f x + e\right )^{4} - 12 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5}\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{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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