Optimal. Leaf size=140 \[ \frac {4 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{105 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}+\frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{21 a f} \]
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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} \[ \frac {4 c^2 \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{105 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}+\frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{21 a f} \]
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))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac {\int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{5/2} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}+\frac {4 \int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2} \, dx}{7 a}\\ &=\frac {2 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{21 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}+\frac {(4 c) \int (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)} \, dx}{21 a}\\ &=\frac {4 c^2 \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{105 a f \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{21 a f}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2}}{7 a f}\\ \end {align*}
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Mathematica [A] time = 1.28, size = 115, normalized size = 0.82 \[ \frac {a^3 c \sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (4725 \sin (e+f x)+665 \sin (3 (e+f x))+21 \sin (5 (e+f x))-15 \sin (7 (e+f x))-1050 \cos (2 (e+f x))-420 \cos (4 (e+f x))-70 \cos (6 (e+f x)))}{6720 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 115, normalized size = 0.82 \[ -\frac {{\left (35 \, a^{3} c \cos \left (f x + e\right )^{6} - 35 \, a^{3} c + {\left (15 \, a^{3} c \cos \left (f x + e\right )^{6} - 24 \, a^{3} c \cos \left (f x + e\right )^{4} - 32 \, a^{3} c \cos \left (f x + e\right )^{2} - 64 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, f \cos \left (f x + e\right )} \]
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 = 0.40, size = 133, normalized size = 0.95 \[ -\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \left (-15 \left (\cos ^{8}\left (f x +e \right )\right )+5 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )-16 \left (\cos ^{6}\left (f x +e \right )\right )+13 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-16 \left (\cos ^{4}\left (f x +e \right )\right )+29 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+58 \sin \left (f x +e \right )-58\right )}{105 f \cos \left (f x +e \right )^{7}} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.14, size = 319, normalized size = 2.28 \[ -\frac {{\mathrm {e}}^{-e\,7{}\mathrm {i}-f\,x\,7{}\mathrm {i}}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {5\,a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{16\,f}+\frac {a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{8\,f}+\frac {a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{48\,f}-\frac {19\,a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{96\,f}-\frac {a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{160\,f}+\frac {a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{224\,f}-\frac {45\,a^3\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}\right )}{2\,\cos \left (e+f\,x\right )} \]
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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