Optimal. Leaf size=238 \[ \frac{16 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{32 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{c f (c-c \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 0.753604, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2841, 2739, 2740, 2737, 2667, 31} \[ \frac{16 a^3 \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{32 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{c f (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2739
Rule 2740
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx &=\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}-\frac{4 \int \frac{(a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{(8 a) \int \frac{(a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{4 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{\left (16 a^2\right ) \int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{16 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{\left (32 a^3\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{16 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}-\frac{\left (32 a^4 \cos (e+f x)\right ) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c \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))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{16 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{\left (32 a^4 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^2 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))^{7/2}}{c f (c-c \sin (e+f x))^{3/2}}+\frac{32 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^2 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{16 a^3 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^2 f \sqrt{c-c \sin (e+f x)}}+\frac{4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.82273, size = 196, normalized size = 0.82 \[ -\frac{a^3 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (-396 \sin (e+f x)-16 \sin (3 (e+f x))-172 \cos (2 (e+f x))+\cos (4 (e+f x))-1536 \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+1536 \sin (e+f x) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-177\right )}{24 c^2 f (\sin (e+f x)-1)^2 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.241, size = 307, normalized size = 1.3 \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 ( \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-4\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8\,\sin \left ( fx+e \right ) -4\,\cos \left ( fx+e \right ) +8 \right ) } \left ( - \left ( \cos \left ( fx+e \right ) \right ) ^{4}+8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +96\,\sin \left ( fx+e \right ) \ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) -192\,\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 ) +44\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+91\,\sin \left ( fx+e \right ) -96\,\ln \left ( 2\, \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1} \right ) +192\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -43 \right ) \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}} \left ( -c \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{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\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{{\left (3 \, a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{2} +{\left (a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{2}\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}}{3 \, c^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{3} -{\left (c^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{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{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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