Optimal. Leaf size=145 \[ -\frac{a 2^{\frac{p}{2}+\frac{3}{2}} (\sin (e+f x)+1)^{\frac{1}{2} (-p-1)} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{p+1}{2};\frac{1}{2} (-p-1),-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f g (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.165132, antiderivative size = 151, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2868, 139, 138} \[ -\frac{a g 2^{\frac{p+3}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac{1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{p+1}{2};\frac{1}{2} (-p-1),-n;\frac{p+3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2868
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx &=\frac{\left (a g (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac{1-p}{2}} (1+\sin (e+f x))^{\frac{1-p}{2}}\right ) \operatorname{Subst}\left (\int (1-x)^{\frac{1}{2} (-1+p)} (1+x)^{\frac{1+p}{2}} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\left (a g (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac{1-p}{2}} (1+\sin (e+f x))^{\frac{1-p}{2}} (c+d \sin (e+f x))^n \left (-\frac{c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname{Subst}\left (\int (1-x)^{\frac{1}{2} (-1+p)} (1+x)^{\frac{1+p}{2}} \left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{2^{\frac{3+p}{2}} a g F_1\left (\frac{1+p}{2};\frac{1}{2} (-1-p),-n;\frac{3+p}{2};\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)) (1+\sin (e+f x))^{\frac{1-p}{2}} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{f (1+p)}\\ \end{align*}
Mathematica [F] time = 3.82345, size = 0, normalized size = 0. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.394, 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 ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )} \left (g \cos \left (f x + e\right )\right )^{p}{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )} \left (g \cos \left (f x + e\right )\right )^{p}{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )} \left (g \cos \left (f x + e\right )\right )^{p}{\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.
[In]
[Out]