Integrand size = 30, antiderivative size = 188 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {2 c^4 \cos (e+f x) \log (1+\sin (e+f x))}{3 f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f \sqrt {3+3 \sin (e+f x)}}+\frac {c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f (3+3 \sin (e+f x))^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (3+3 \sin (e+f x))^{5/2}} \]
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Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2818, 2819, 2816, 2746, 31} \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {6 c^4 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {3 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a \sin (e+f x)+a}}+\frac {3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a \sin (e+f x)+a)^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a \sin (e+f x)+a)^{5/2}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2818
Rule 2819
Rubi steps \begin{align*} \text {integral}& = -\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c) \int \frac {(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{2 a} \\ & = \frac {3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a+a \sin (e+f x))^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac {\left (3 c^2\right ) \int \frac {(c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2} \\ & = \frac {3 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a+a \sin (e+f x))^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac {\left (6 c^3\right ) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2} \\ & = \frac {3 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a+a \sin (e+f x))^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac {\left (6 c^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{a \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {3 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a+a \sin (e+f x))^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a+a \sin (e+f x))^{5/2}}+\frac {\left (6 c^4 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {6 c^4 \cos (e+f x) \log (1+\sin (e+f x))}{a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {3 c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {3 c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 a f (a+a \sin (e+f x))^{3/2}}-\frac {c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{2 f (a+a \sin (e+f x))^{5/2}} \\ \end{align*}
Time = 7.66 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.01 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} \left (28+\cos (2 (e+f x)) \left (4-24 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+72 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\left (41+96 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin (e+f x)+\sin (3 (e+f x))\right )}{36 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2}} \]
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Time = 2.99 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {\sec \left (f x +e \right ) \left (\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+6 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-12 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+10 \left (\cos ^{2}\left (f x +e \right )\right )-12 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )+24 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-6 \sin \left (f x +e \right )-12 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+24 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-10\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}{f \left (\sin \left (f x +e \right )+1\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{2}}\) | \(217\) |
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\[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.82 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {a} c^{\frac {7}{2}} {\left (\frac {2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {6 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]
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Timed out. \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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