Optimal. Leaf size=176 \[ -\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f} \]
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Rubi [A] time = 0.419948, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {2736, 2674, 2673} \[ -\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \frac{(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx &=\frac{\int \sec ^4(e+f x) (c-c \sin (e+f x))^{13/2} \, dx}{a^2 c^2}\\ &=\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{16 \int \sec ^4(e+f x) (c-c \sin (e+f x))^{11/2} \, dx}{5 a^2 c}\\ &=\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{64 \int \sec ^4(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{5 a^2}\\ &=\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}+\frac{(512 c) \int \sec ^4(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{5 a^2}\\ &=-\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}-\frac{\left (2048 c^2\right ) \int \sec ^4(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{5 a^2}\\ &=\frac{4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}-\frac{1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac{128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}+\frac{32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac{2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f}\\ \end{align*}
Mathematica [A] time = 3.07125, size = 124, normalized size = 0.7 \[ \frac{c^4 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (8568 \sin (e+f x)+56 \sin (3 (e+f x))-1044 \cos (2 (e+f x))+3 \cos (4 (e+f x))+6825)}{60 a^2 f (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.638, size = 91, normalized size = 0.5 \begin{align*} -{\frac{2\,{c}^{5} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 3\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}-28\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}+258\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}+1092\,\sin \left ( fx+e \right ) +723 \right ) }{15\,{a}^{2} \left ( 1+\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) f}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.73688, size = 513, normalized size = 2.91 \begin{align*} -\frac{2 \,{\left (723 \, c^{\frac{9}{2}} + \frac{2184 \, c^{\frac{9}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{5370 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{10696 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{15021 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{21168 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac{20748 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{21168 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac{15021 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{10696 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac{5370 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac{2184 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + \frac{723 \, c^{\frac{9}{2}} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )}}{15 \,{\left (a^{2} + \frac{3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} f{\left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06822, size = 261, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (3 \, c^{4} \cos \left (f x + e\right )^{4} - 264 \, c^{4} \cos \left (f x + e\right )^{2} + 984 \, c^{4} + 28 \,{\left (c^{4} \cos \left (f x + e\right )^{2} + 38 \, c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{15 \,{\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\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 [B] time = 2.09839, size = 887, normalized size = 5.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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