Integrand size = 35, antiderivative size = 308 \[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=-\frac {\operatorname {AppellF1}\left (1+p,\frac {1+p}{2},\frac {1+p}{2},2+p,\frac {a+b}{a+b \sin (e+f x)},\frac {a-b}{a+b \sin (e+f x)}\right ) \sec (e+f x) (g \sec (e+f x))^p \left (-\frac {b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac {1+p}{2}} \left (\frac {b (1+\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac {1+p}{2}}}{(b c-a d) f (1+p)}+\frac {\operatorname {AppellF1}\left (1+p,\frac {1+p}{2},\frac {1+p}{2},2+p,\frac {c+d}{c+d \sin (e+f x)},\frac {c-d}{c+d \sin (e+f x)}\right ) \sec (e+f x) (g \sec (e+f x))^p \left (-\frac {d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1+p}{2}} \left (\frac {d (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1+p}{2}}}{(b c-a d) f (1+p)} \]
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Time = 0.40 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3005, 3003, 2782} \[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\frac {\sec (e+f x) (g \sec (e+f x))^p \left (-\frac {d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {p+1}{2}} \left (\frac {d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac {p+1}{2}} \operatorname {AppellF1}\left (p+1,\frac {p+1}{2},\frac {p+1}{2},p+2,\frac {c+d}{c+d \sin (e+f x)},\frac {c-d}{c+d \sin (e+f x)}\right )}{f (p+1) (b c-a d)}-\frac {\sec (e+f x) (g \sec (e+f x))^p \left (-\frac {b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac {p+1}{2}} \left (\frac {b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right )^{\frac {p+1}{2}} \operatorname {AppellF1}\left (p+1,\frac {p+1}{2},\frac {p+1}{2},p+2,\frac {a+b}{a+b \sin (e+f x)},\frac {a-b}{a+b \sin (e+f x)}\right )}{f (p+1) (b c-a d)} \]
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Rule 2782
Rule 3003
Rule 3005
Rubi steps \begin{align*} \text {integral}& = \left ((g \cos (e+f x))^p (g \sec (e+f x))^p\right ) \int \frac {(g \cos (e+f x))^{-p}}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx \\ & = \left ((g \cos (e+f x))^p (g \sec (e+f x))^p\right ) \int \left (\frac {b (g \cos (e+f x))^{-p}}{(b c-a d) (a+b \sin (e+f x))}-\frac {d (g \cos (e+f x))^{-p}}{(b c-a d) (c+d \sin (e+f x))}\right ) \, dx \\ & = \frac {\left (b (g \cos (e+f x))^p (g \sec (e+f x))^p\right ) \int \frac {(g \cos (e+f x))^{-p}}{a+b \sin (e+f x)} \, dx}{b c-a d}-\frac {\left (d (g \cos (e+f x))^p (g \sec (e+f x))^p\right ) \int \frac {(g \cos (e+f x))^{-p}}{c+d \sin (e+f x)} \, dx}{b c-a d} \\ & = -\frac {\operatorname {AppellF1}\left (1+p,\frac {1+p}{2},\frac {1+p}{2},2+p,\frac {a+b}{a+b \sin (e+f x)},\frac {a-b}{a+b \sin (e+f x)}\right ) \sec (e+f x) (g \sec (e+f x))^p \left (-\frac {b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac {1+p}{2}} \left (\frac {b (1+\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac {1+p}{2}}}{(b c-a d) f (1+p)}+\frac {\operatorname {AppellF1}\left (1+p,\frac {1+p}{2},\frac {1+p}{2},2+p,\frac {c+d}{c+d \sin (e+f x)},\frac {c-d}{c+d \sin (e+f x)}\right ) \sec (e+f x) (g \sec (e+f x))^p \left (-\frac {d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1+p}{2}} \left (\frac {d (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1+p}{2}}}{(b c-a d) f (1+p)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(5113\) vs. \(2(308)=616\).
Time = 26.15 (sec) , antiderivative size = 5113, normalized size of antiderivative = 16.60 \[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (g \sec \left (f x +e \right )\right )^{p}}{\left (a +b \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )}d x\]
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\[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{p}}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}} \,d x } \]
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\[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{p}}{\left (a + b \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )}\, dx \]
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\[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{p}}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}} \,d x } \]
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\[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\int { \frac {\left (g \sec \left (f x + e\right )\right )^{p}}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(g \sec (e+f x))^p}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx=\int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^p}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \]
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