Integrand size = 30, antiderivative size = 169 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {162 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt {3+3 \sin (e+f x)}}-\frac {108 \cos (e+f x) \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac {9 \cos (e+f x) (3+3 \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac {3 \cos (e+f x) (3+3 \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \]
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Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817} \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac {a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \]
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Rule 2817
Rule 2819
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac {1}{7} (6 a) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2} \, dx \\ & = -\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac {1}{7} \left (4 a^2\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2} \, dx \\ & = -\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f}+\frac {1}{35} \left (8 a^3\right ) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx \\ & = -\frac {2 a^4 \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}{35 f}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{7 f}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{7 f} \\ \end{align*}
Time = 1.48 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {27 \sqrt {3} c^3 \sec ^6(e+f x) (-1+\sin (e+f x))^3 (1+\sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \left (-35+35 \sin ^2(e+f x)-21 \sin ^4(e+f x)+5 \sin ^6(e+f x)\right ) \tan (e+f x)}{35 f} \]
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Time = 4.84 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.44
method | result | size |
default | \(\frac {\tan \left (f x +e \right ) a^{3} c^{3} \left (5 \left (\cos ^{6}\left (f x +e \right )\right )+6 \left (\cos ^{4}\left (f x +e \right )\right )+8 \left (\cos ^{2}\left (f x +e \right )\right )+16\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{35 f}\) | \(75\) |
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Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.60 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {{\left (5 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 6 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 8 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{3} c^{3}\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{35 \, f \cos \left (f x + e\right )} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.21 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=-\frac {32 \, {\left (20 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 70 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} + 84 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 35 \, a^{3} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}\right )} \sqrt {a} \sqrt {c}}{35 \, f} \]
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Time = 11.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.06 \[ \int (3+3 \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{7/2} \, dx=\frac {\frac {1225\,a^3\,c^3\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{32}+\frac {245\,a^3\,c^3\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{32}+\frac {49\,a^3\,c^3\,\sin \left (5\,e+5\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{32}+\frac {5\,a^3\,c^3\,\sin \left (7\,e+7\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{32}}{70\,f\,\cos \left (e+f\,x\right )} \]
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