Integrand size = 23, antiderivative size = 125 \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {7}{3},-\frac {1}{2},-\frac {1}{2},\frac {10}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \]
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Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2783, 143} \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\frac {3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} \operatorname {AppellF1}\left (\frac {7}{3},-\frac {1}{2},-\frac {1}{2},\frac {10}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right )}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rule 143
Rule 2783
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \text {Subst}\left (\int (c+d x)^{4/3} \sqrt {-\frac {d}{c-d}-\frac {d x}{c-d}} \sqrt {\frac {d}{c+d}-\frac {d x}{c+d}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \\ & = \frac {3 \operatorname {AppellF1}\left (\frac {7}{3},-\frac {1}{2},-\frac {1}{2},\frac {10}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(125)=250\).
Time = 1.56 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.41 \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=-\frac {3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (12 \left (4 c^4+3 c^2 d^2-7 d^4\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}}-3 c \left (4 c^2+51 d^2\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sin (e+f x))}{c-d}} (c+d \sin (e+f x))+4 d^2 \cos ^2(e+f x) \left (-4 c^2+7 d^2+14 d^2 \cos (2 (e+f x))-44 c d \sin (e+f x)\right )\right )}{1120 d^3 f} \]
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\[\int \left (\cos ^{2}\left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}d x\]
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\[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
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\[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\int \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {4}{3}} \cos ^{2}{\left (e + f x \right )}\, dx \]
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\[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
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\[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]
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