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 = 0.62, antiderivative size = 308, normalized size of antiderivative = 1.00, 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 2703
Rule 2924
Rule 2926
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*}
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Mathematica [B] time = 30.05, size = 5113, normalized size = 16.60 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (g \sec \left (f x + e\right )\right )^{p}}{b d \cos \left (f x + e\right )^{2} - a c - b d - {\left (b c + a d\right )} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \sec \left (f x + e\right )\right )^{p}}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \sec \left (f x +e \right )\right )^{p}}{\left (a +b \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \sec \left (f x + e\right )\right )^{p}}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^p}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \sec {\left (e + f x \right )}\right )^{p}}{\left (a + b \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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