Optimal. Leaf size=237 \[ \frac{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{24 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a \sin (e+f x)+a)^{5/2}} \]
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Rubi [A] time = 0.492747, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2739, 2740, 2737, 2667, 31} \[ \frac{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a \sin (e+f x)+a}}+\frac{24 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a \sin (e+f x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2739
Rule 2740
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}-\frac{(2 c) \int \frac{(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{a}\\ &=\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (6 c^2\right ) \int \frac{(c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a^2}\\ &=\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (12 c^3\right ) \int \frac{(c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a^2}\\ &=\frac{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (24 c^4\right ) \int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a^2}\\ &=\frac{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (24 c^5 \cos (e+f x)\right ) \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{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (24 c^5 \cos (e+f x)\right ) \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{24 c^5 \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{12 c^4 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{2 f (a+a \sin (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 5.14368, size = 202, normalized size = 0.85 \[ \frac{c^4 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (320 \sin (e+f x)+24 \sin (3 (e+f x))+\cos (4 (e+f x))+\cos (2 (e+f x)) \left (106-384 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )+1152 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+1536 \sin (e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+273\right )}{16 f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.196, size = 685, normalized size = 2.9 \begin{align*} \text{result too large to display} \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 (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{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{{\left (c^{4} \cos \left (f x + e\right )^{4} - 8 \, c^{4} \cos \left (f x + e\right )^{2} + 8 \, c^{4} + 4 \,{\left (c^{4} \cos \left (f x + e\right )^{2} - 2 \, c^{4}\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 \, 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{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{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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