Integrand size = 38, antiderivative size = 145 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}} \]
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Time = 0.42 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2822, 2821} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}} \]
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Rule 2821
Rule 2822
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{15/2}} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{7 a c^2} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{84 a c^3} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(419\) vs. \(2(145)=290\).
Time = 12.87 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.89 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {16 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {16 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {24 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}} \]
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Time = 0.39 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \left (9 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-63 \left (\cos ^{5}\left (f x +e \right )\right )-216 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+469 \left (\cos ^{3}\left (f x +e \right )\right )+790 \cos \left (f x +e \right ) \sin \left (f x +e \right )-854 \cos \left (f x +e \right )-688 \tan \left (f x +e \right )+448 \sec \left (f x +e \right )\right )}{105 f \left (\cos ^{6}\left (f x +e \right )+6 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-18 \left (\cos ^{4}\left (f x +e \right )\right )-32 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{2}\left (f x +e \right )\right )+32 \sin \left (f x +e \right )-32\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{8}}\) | \(200\) |
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Time = 0.34 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=-\frac {{\left (35 \, a^{3} \cos \left (f x + e\right )^{4} - 154 \, a^{3} \cos \left (f x + e\right )^{2} + 128 \, a^{3} - 14 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 8 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, {\left (7 \, c^{9} f \cos \left (f x + e\right )^{7} - 56 \, c^{9} f \cos \left (f x + e\right )^{5} + 112 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right ) - {\left (c^{9} f \cos \left (f x + e\right )^{7} - 24 \, c^{9} f \cos \left (f x + e\right )^{5} + 80 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.39 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=-\frac {{\left (35 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 105 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 126 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 70 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{840 \, c^{9} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14}} \]
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Time = 15.98 (sec) , antiderivative size = 764, normalized size of antiderivative = 5.27 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Too large to display} \]
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