\(\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx\) [927]

Optimal result

Integrand size = 27, antiderivative size = 54 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(1,2+m+n,2+m,1+\sin (c+d x)) \sin ^{1+n}(c+d x) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]

[Out]

-hypergeom([1, 2+m+n],[2+m],1+sin(d*x+c))*sin(d*x+c)^(1+n)*(a+a*sin(d*x+c))^(1+m)/a/d/(1+m)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 68, 66} \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {(\sin (c+d x)+1)^{-m} \sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^m \operatorname {Hypergeometric2F1}(-m,n+1,n+2,-\sin (c+d x))}{d (n+1)} \]

[In]

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

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

Rule 2912

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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {x}{a}\right )^n (a+x)^m \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\left ((1+\sin (c+d x))^{-m} (a+a \sin (c+d x))^m\right ) \text {Subst}\left (\int \left (\frac {x}{a}\right )^n \left (1+\frac {x}{a}\right )^m \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\operatorname {Hypergeometric2F1}(-m,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x) (1+\sin (c+d x))^{-m} (a+a \sin (c+d x))^m}{d (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}(-m,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x) (1+\sin (c+d x))^{-m} (a+a \sin (c+d x))^m}{d (1+n)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \sin ^{n}{\left (c + d x \right )} \cos {\left (c + d x \right )}\, dx \]

[In]

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

[Out]

Integral((a*(sin(c + d*x) + 1))**m*sin(c + d*x)**n*cos(c + d*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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