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

Optimal. Leaf size=54 \[ -\frac{\sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^{m+1} \, _2F_1(1,m+n+2;m+2;\sin (c+d x)+1)}{a d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0778548, antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 66, 64} \[ \frac{(\sin (c+d x)+1)^{-m} \sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^m \, _2F_1(-m,n+1;n+2;-\sin (c+d x))}{d (n+1)} \]

Antiderivative was successfully verified.

[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 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 66

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 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), 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])))

Rubi steps

\begin{align*} \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac{\operatorname{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 ) \operatorname{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{\, _2F_1(-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]  time = 0.0799043, size = 61, normalized size = 1.13 \[ \frac{(\sin (c+d x)+1)^{-m} \sin ^{n+1}(c+d x) (a \sin (c+d x)+a)^m \, _2F_1(-m,n+1;n+2;-\sin (c+d x))}{d (n+1)} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 3.917, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)