\(\int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx\) [1044]

Optimal result

Integrand size = 33, antiderivative size = 145 \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=-\frac {2^{\frac {3}{2}+\frac {p}{2}} a \operatorname {AppellF1}\left (\frac {1+p}{2},\frac {1}{2} (-1-p),-n,\frac {3+p}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) (g \cos (e+f x))^{1+p} (1+\sin (e+f x))^{\frac {1}{2} (-1-p)} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{f g (1+p)} \]

[Out]

-2^(3/2+1/2*p)*a*AppellF1(1/2+1/2*p,-n,-1/2-1/2*p,3/2+1/2*p,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))*(g*cos(
f*x+e))^(p+1)*(1+sin(f*x+e))^(-1/2-1/2*p)*(c+d*sin(f*x+e))^n/f/g/(p+1)/(((c+d*sin(f*x+e))/(c+d))^n)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2947, 144, 143} \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=-\frac {a g 2^{\frac {p+3}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac {1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} \operatorname {AppellF1}\left (\frac {p+1}{2},\frac {1}{2} (-p-1),-n,\frac {p+3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{f (p+1)} \]

[In]

Int[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]

[Out]

-((2^((3 + p)/2)*a*g*AppellF1[(1 + p)/2, (-1 - p)/2, -n, (3 + p)/2, (1 - Sin[e + f*x])/2, (d*(1 - Sin[e + f*x]
))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n
)/(f*(1 + p)*((c + d*Sin[e + f*x])/(c + d))^n))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 2947

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)]), x_Symbol] :> Dist[c*g*((g*Cos[e + f*x])^(p - 1)/(f*(1 + Sin[e + f*x])^((p - 1)/2)*(1 - Sin[e +
f*x])^((p - 1)/2))), Subst[Int[(1 + (d/c)*x)^((p + 1)/2)*(1 - (d/c)*x)^((p - 1)/2)*(a + b*x)^m, x], x, Sin[e +
f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && NeQ[a^2 - b^2, 0] && EqQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \operatorname {AppellF1}\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]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]

[In]

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n,x]

[Out]

Integrate[(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]

Maple [F]

\[\int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]

[In]

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x)

[Out]

int((g*cos(f*x+e))^p*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x)

Fricas [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e) + a)*(g*cos(f*x + e))^p*(d*sin(f*x + e) + c)^n, x)

Sympy [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=a \left (\int \left (g \cos {\left (e + f x \right )}\right )^{p} \left (c + d \sin {\left (e + f x \right )}\right )^{n}\, dx + \int \left (g \cos {\left (e + f x \right )}\right )^{p} \left (c + d \sin {\left (e + f x \right )}\right )^{n} \sin {\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((g*cos(f*x+e))**p*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))**n,x)

[Out]

a*(Integral((g*cos(e + f*x))**p*(c + d*sin(e + f*x))**n, x) + Integral((g*cos(e + f*x))**p*(c + d*sin(e + f*x)
)**n*sin(e + f*x), x))

Maxima [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)*(g*cos(f*x + e))^p*(d*sin(f*x + e) + c)^n, x)

Giac [F]

\[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (g \cos \left (f x + e\right )\right )^{p} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

integrate((g*cos(f*x+e))^p*(a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)*(g*cos(f*x + e))^p*(d*sin(f*x + e) + c)^n, x)

Mupad [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^p (a+a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^p\,\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]

[In]

int((g*cos(e + f*x))^p*(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^n,x)

[Out]

int((g*cos(e + f*x))^p*(a + a*sin(e + f*x))*(c + d*sin(e + f*x))^n, x)