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]

-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)

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Rubi [A]  time = 0.08, 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 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])))

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 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]

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*}

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Mathematica [A]  time = 0.07, 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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right ), x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )\,{d x} \]

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)

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maple [F]  time = 5.19, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \sin ^{n}{\left (c + d x \right )} \cos {\left (c + d x \right )}\, dx \]

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]

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

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