Optimal. Leaf size=168 \[ \frac{g 2^{\frac{p+1}{2}} (1-\sin (e+f x))^{\frac{1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (\frac{1}{2} (2 m+p+1);\frac{1-p}{2},-n;\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (2 m+p+1)} \]
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Rubi [A] time = 0.28029, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2921, 140, 139, 138} \[ \frac{g 2^{\frac{p+1}{2}} (1-\sin (e+f x))^{\frac{1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (\frac{1}{2} (2 m+p+1);\frac{1-p}{2},-n;\frac{1}{2} (2 m+p+3);\frac{1}{2} (\sin (e+f x)+1),-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (2 m+p+1)} \]
Antiderivative was successfully verified.
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Rule 2921
Rule 140
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx &=\frac{\left (g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{\frac{1-p}{2}} (a+a \sin (e+f x))^{\frac{1-p}{2}}\right ) \operatorname{Subst}\left (\int (a-a x)^{\frac{1}{2} (-1+p)} (a+a x)^{m+\frac{1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (2^{-\frac{1}{2}+\frac{p}{2}} g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{-\frac{1}{2}+\frac{1-p}{2}+\frac{p}{2}} \left (\frac{a-a \sin (e+f x)}{a}\right )^{\frac{1}{2}-\frac{p}{2}} (a+a \sin (e+f x))^{\frac{1-p}{2}}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{\frac{1}{2} (-1+p)} (a+a x)^{m+\frac{1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (2^{-\frac{1}{2}+\frac{p}{2}} g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{-\frac{1}{2}+\frac{1-p}{2}+\frac{p}{2}} \left (\frac{a-a \sin (e+f x)}{a}\right )^{\frac{1}{2}-\frac{p}{2}} (a+a \sin (e+f x))^{\frac{1-p}{2}} (c+d \sin (e+f x))^n \left (\frac{a (c+d \sin (e+f x))}{a c-a d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{\frac{1}{2} (-1+p)} (a+a x)^{m+\frac{1}{2} (-1+p)} \left (\frac{a c}{a c-a d}+\frac{a d x}{a c-a d}\right )^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{2^{\frac{1+p}{2}} g F_1\left (\frac{1}{2} (1+2 m+p);\frac{1-p}{2},-n;\frac{1}{2} (3+2 m+p);\frac{1}{2} (1+\sin (e+f x)),-\frac{d (1+\sin (e+f x))}{c-d}\right ) (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac{1-p}{2}} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+2 m+p)}\\ \end{align*}
Mathematica [B] time = 8.34151, size = 798, normalized size = 4.75 \[ -\frac{2 F_1\left (\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) (g \cos (e+f x))^p \cos \left (\frac{1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^n \sin \left (\frac{1}{4} (2 e+2 f x+\pi )\right )}{f \left (-\frac{d n F_1\left (\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \cos ^2(e+f x)}{c+d \sin (e+f x)}+\frac{2 (p+1) \left ((c-d) n F_1\left (\frac{p+3}{2};m+n+p+1,1-n;\frac{p+5}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right )-(c+d) (m+n+p+1) F_1\left (\frac{p+3}{2};m+n+p+2,-n;\frac{p+5}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right )\right ) \cot ^2\left (\frac{1}{4} (2 e+2 f x+\pi )\right )}{(c+d) (p+3)}+2 (n+p) F_1\left (\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \sin ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )-\frac{2 (c-d) n F_1\left (\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \sin ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d \sin (e+f x)}+F_1\left (\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right )+p F_1\left (\frac{p+1}{2};m+n+p+1,-n;\frac{p+3}{2};-\tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right ),-\frac{(c-d) \tan ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \sin (e+f x)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 2.763, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{p} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \cos \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \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 (\left (g \cos \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (g \cos \left (f x + e\right )\right )^{p}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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