Optimal. Leaf size=145 \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}} \]
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Rubi [A] time = 0.538983, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2841, 2743, 2742} \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}} \]
Antiderivative was successfully verified.
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Rule 2841
Rule 2743
Rule 2742
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))^{17/2}} \, dx &=\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{15/2}} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{7 a c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{84 a c^3}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}\\ \end{align*}
Mathematica [B] time = 6.83592, size = 419, normalized size = 2.89 \[ \frac{(a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}{3 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{2 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{24 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}{5 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{3 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}{7 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.251, size = 243, normalized size = 1.7 \begin{align*}{\frac{ \left ( 9\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+63\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-216\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-406\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +790\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+448\,\sin \left ( fx+e \right ) -688 \right ) \sin \left ( fx+e \right ) \left ( \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 \right ) }{105\,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 ( 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{17}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.99511, size = 517, normalized size = 3.57 \begin{align*} -\frac{{\left (35 \, a^{3} \cos \left (f x + e\right )^{4} - 154 \, a^{3} \cos \left (f x + e\right )^{2} + 128 \, a^{3} - 14 \,{\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 8 \, a^{3}\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}}{105 \,{\left (7 \, c^{9} f \cos \left (f x + e\right )^{7} - 56 \, c^{9} f \cos \left (f x + e\right )^{5} + 112 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right ) -{\left (c^{9} f \cos \left (f x + e\right )^{7} - 24 \, c^{9} f \cos \left (f x + e\right )^{5} + 80 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \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 [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{17}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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