Optimal. Leaf size=239 \[ \frac {16 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {8 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a \sin (e+f x)+a}}+\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a \sin (e+f x)+a}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a \sin (e+f x)+a}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.74, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2841, 2740, 2737, 2667, 31} \[ \frac {8 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a \sin (e+f x)+a}}+\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a \sin (e+f x)+a}}+\frac {16 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a \sin (e+f x)+a}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rule 2737
Rule 2740
Rule 2841
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac {\int \frac {(c-c \sin (e+f x))^{9/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a c}\\ &=\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {(c-c \sin (e+f x))^{7/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {(4 c) \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (8 c^2\right ) \int \frac {(c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac {8 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (16 c^3\right ) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac {8 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (16 c^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {8 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {\left (16 c^4 \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {16 c^4 \cos (e+f x) \log (1+\sin (e+f x))}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {8 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a f \sqrt {a+a \sin (e+f x)}}+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.47, size = 471, normalized size = 1.97 \[ \frac {5 \sin (3 (e+f x)) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{12 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {23 \cos (2 (e+f x)) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{8 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {\cos (4 (e+f x)) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{32 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {65 \sin (e+f x) (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{4 f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {32 (c-c \sin (e+f x))^{7/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f (a (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, c^{3} \cos \left (f x + e\right )^{4} - 4 \, c^{3} \cos \left (f x + e\right )^{2} - {\left (c^{3} \cos \left (f x + e\right )^{4} - 4 \, c^{3} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 = 0.40, size = 252, normalized size = 1.05 \[ \frac {\left (3 \left (\cos ^{4}\left (f x +e \right )\right )+20 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-72 \left (\cos ^{2}\left (f x +e \right )\right )-192 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-200 \sin \left (f x +e \right )+384 \ln \left (-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+69\right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {7}{2}} \left (\cos ^{2}\left (f x +e \right )+\sin \left (f x +e \right ) \cos \left (f x +e \right )+\cos \left (f x +e \right )-2 \sin \left (f x +e \right )-2\right )}{12 f \left (\cos ^{4}\left (f x +e \right )-\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+3 \left (\cos ^{3}\left (f x +e \right )\right )+4 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-8 \left (\cos ^{2}\left (f x +e \right )\right )+4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-4 \cos \left (f x +e \right )-8 \sin \left (f x +e \right )+8\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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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