3.912 \(\int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx\)

Optimal. Leaf size=88 \[ \frac{(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \]

[Out]

(Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Si
n[e + f*x])^n)/(a*f*(1 + m)*((c + d*Sin[e + f*x])/(c - d))^n)

________________________________________________________________________________________

Rubi [A]  time = 0.13977, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2833, 70, 69} \[ \frac{(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Si
n[e + f*x])^n)/(a*f*(1 + m)*((c + d*Sin[e + f*x])/(c - d))^n)

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^m \left (c+\frac{d x}{a}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\left ((c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n}\right ) \operatorname{Subst}\left (\int (a+x)^m \left (\frac{c}{c-d}+\frac{d x}{a (c-d)}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\, _2F_1\left (1+m,-n;2+m;-\frac{d (1+\sin (e+f x))}{c-d}\right ) (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+m)}\\ \end{align*}

Mathematica [A]  time = 0.140462, size = 88, normalized size = 1. \[ \frac{(a \sin (e+f x)+a)^{m+1} (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Si
n[e + f*x])^n)/(a*f*(1 + m)*((c + d*Sin[e + f*x])/(c - d))^n)

________________________________________________________________________________________

Maple [F]  time = 0.33, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( fx+e \right ) \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.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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