Optimal. Leaf size=140 \[ \frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)}}{15 a f} \]
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Rubi [A] time = 0.52, antiderivative size = 140, normalized size of antiderivative = 1.00, 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, 2740, 2738} \[ \frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{15 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)}}{15 a f} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2740
Rule 2841
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac {\int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac {2 \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx}{3 a}\\ &=\frac {2 c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{15 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}+\frac {(4 c) \int (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx}{15 a}\\ &=\frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{15 a f \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{15 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{6 a f}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 152, normalized size = 1.09 \[ -\frac {c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt {c-c \sin (e+f x)} (600 \sin (e+f x)+100 \sin (3 (e+f x))+12 \sin (5 (e+f x))-75 \cos (2 (e+f x))-30 \cos (4 (e+f x))-5 \cos (6 (e+f x)))}{960 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 102, normalized size = 0.73 \[ -\frac {{\left (5 \, a^{2} c \cos \left (f x + e\right )^{6} - 5 \, a^{2} c - 2 \, {\left (3 \, a^{2} c \cos \left (f x + e\right )^{4} + 4 \, a^{2} c \cos \left (f x + e\right )^{2} + 8 \, a^{2} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 116, normalized size = 0.83 \[ -\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \left (-5 \left (\cos ^{6}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-6 \left (\cos ^{4}\left (f x +e \right )\right )+3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-8 \left (\cos ^{2}\left (f x +e \right )\right )+11 \sin \left (f x +e \right )-11\right )}{30 f \cos \left (f x +e \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.52, size = 122, normalized size = 0.87 \[ -\frac {a^2\,c\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (75\,\cos \left (e+f\,x\right )+105\,\cos \left (3\,e+3\,f\,x\right )+35\,\cos \left (5\,e+5\,f\,x\right )+5\,\cos \left (7\,e+7\,f\,x\right )-700\,\sin \left (2\,e+2\,f\,x\right )-112\,\sin \left (4\,e+4\,f\,x\right )-12\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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