Optimal. Leaf size=308 \[ \frac{\sec (e+f x) (g \sec (e+f x))^p \left (-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac{p+1}{2}} \left (\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac{p+1}{2}} F_1\left (p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right )}{f (p+1) (b c-a d)}-\frac{\sec (e+f x) (g \sec (e+f x))^p \left (-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac{p+1}{2}} \left (\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right )^{\frac{p+1}{2}} F_1\left (p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right )}{f (p+1) (b c-a d)} \]
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Rubi [A] time = 0.615735, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {2926, 2924, 2703} \[ \frac{\sec (e+f x) (g \sec (e+f x))^p \left (-\frac{d (1-\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac{p+1}{2}} \left (\frac{d (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac{p+1}{2}} F_1\left (p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{c+d}{c+d \sin (e+f x)},\frac{c-d}{c+d \sin (e+f x)}\right )}{f (p+1) (b c-a d)}-\frac{\sec (e+f x) (g \sec (e+f x))^p \left (-\frac{b (1-\sin (e+f x))}{a+b \sin (e+f x)}\right )^{\frac{p+1}{2}} \left (\frac{b (\sin (e+f x)+1)}{a+b \sin (e+f x)}\right )^{\frac{p+1}{2}} F_1\left (p+1;\frac{p+1}{2},\frac{p+1}{2};p+2;\frac{a+b}{a+b \sin (e+f x)},\frac{a-b}{a+b \sin (e+f x)}\right )}{f (p+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2926
Rule 2924
Rule 2703
Rubi steps
\begin{align*} \int \frac{(g \sec (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 \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{F_1\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{F_1\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*}
Mathematica [B] time = 28.9841, size = 5101, normalized size = 16.56 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.807, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( g\sec \left ( fx+e \right ) \right ) ^{p}}{ \left ( a+b\sin \left ( fx+e \right ) \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (g \sec \left (f x + e\right )\right )^{p}}{b d \cos \left (f x + e\right )^{2} - a c - b d -{\left (b c + a d\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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