Optimal. Leaf size=241 \[ \frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{c^4 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.78, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2841, 2739, 2740, 2737, 2667, 31} \[ \frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{c^4 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rule 2737
Rule 2739
Rule 2740
Rule 2841
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))^{9/2}} \, dx &=\frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {4 \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{3 c^2}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {(2 a) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c^3}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (4 a^2\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^4}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (8 a^3\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^4}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (8 a^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}+\frac {\left (8 a^4 \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.61, size = 442, normalized size = 1.83 \[ \frac {\sin (e+f x) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}{f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {24 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}{f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{3 f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 3.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{2} + {\left (a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{5 \, c^{5} \cos \left (f x + e\right )^{4} - 20 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5} - {\left (c^{5} \cos \left (f x + e\right )^{4} - 12 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5}\right )} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.41, size = 435, normalized size = 1.80 \[ -\frac {\left (48 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-24 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-3 \left (\cos ^{4}\left (f x +e \right )\right )-49 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-144 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+72 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-192 \sin \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+96 \sin \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+63 \left (\cos ^{2}\left (f x +e \right )\right )+76 \sin \left (f x +e \right )+192 \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-96 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-60\right ) \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )-2 \sin \left (f x +e \right )-\cos \left (f x +e \right )+2\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}}{3 f \left (\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+\cos ^{4}\left (f x +e \right )-4 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \left (\cos ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+8\right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {9}{2}}} \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]
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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