Optimal. Leaf size=143 \[ \frac{c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{5/2}} \]
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Rubi [A] time = 0.302824, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2739, 2737, 2667, 31} \[ \frac{c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{a f (a \sin (e+f x)+a)^{3/2}}-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a \sin (e+f x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2739
Rule 2737
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac{c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (a+a \sin (e+f x))^{5/2}}-\frac{c \int \frac{(c-c \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{a}\\ &=\frac{c^2 \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) (c-c \sin (e+f x))^{3/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{c^2 \int \frac{\sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a^2}\\ &=\frac{c^2 \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) (c-c \sin (e+f x))^{3/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (c^3 \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{c^2 \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) (c-c \sin (e+f x))^{3/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac{\left (c^3 \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{c^3 \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{c^2 \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) (c-c \sin (e+f x))^{3/2}}{2 f (a+a \sin (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.18108, size = 172, normalized size = 1.2 \[ \frac{c^2 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (-\cos (2 (e+f x)) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+3 \log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+4 \sin (e+f x) \left (\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+1\right )+2\right )}{f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 566, normalized size = 4. \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 [A] time = 1.81496, size = 247, normalized size = 1.73 \begin{align*} \frac{\frac{8 \, \sqrt{a} c^{\frac{5}{2}} \sin \left (f x + e\right )^{2}}{{\left (a^{3} + \frac{4 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{6 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{4 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{2 \, c^{\frac{5}{2}} \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{\frac{5}{2}}} + \frac{c^{\frac{5}{2}} \log \left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{a^{\frac{5}{2}}}}{f} \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^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}\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{5}{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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