\(\int \cos ^n(c+d x) \sin (a+b x) \, dx\) [209]

Optimal result

Integrand size = 15, antiderivative size = 277 \[ \int \cos ^n(c+d x) \sin (a+b x) \, dx=-\frac {2^{-1-n} e^{i (a-c n)+i (b-d n) x+i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),-e^{2 i (c+d x)}\right )}{b-d n}-\frac {2^{-1-n} e^{-i (a+c n)-i (b+d n) x+i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^{-n} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},-e^{2 i (c+d x)}\right )}{b+d n} \]

[Out]

-2^(-1-n)*exp(I*(-c*n+a)+I*(-d*n+b)*x+I*n*(d*x+c))*(exp(-I*(d*x+c))+exp(I*(d*x+c)))^n*hypergeom([-n, 1/2*(-d*n
+b)/d],[1+1/2*b/d-1/2*n],-exp(2*I*(d*x+c)))/((1+exp(2*I*c+2*I*d*x))^n)/(-d*n+b)-2^(-1-n)*exp(-I*(c*n+a)-I*(d*n
+b)*x+I*n*(d*x+c))*(exp(-I*(d*x+c))+exp(I*(d*x+c)))^n*hypergeom([-n, 1/2*(-d*n-b)/d],[1+1/2*(-d*n-b)/d],-exp(2
*I*(d*x+c)))/((1+exp(2*I*c+2*I*d*x))^n)/(d*n+b)

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4651, 2323, 2285, 2283} \[ \int \cos ^n(c+d x) \sin (a+b x) \, dx=-\frac {2^{-n-1} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {b-d n}{2 d},\frac {1}{2} \left (\frac {b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac {2^{-n-1} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,-\frac {b+d n}{2 d},1-\frac {b+d n}{2 d},-e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n} \]

[In]

Int[Cos[c + d*x]^n*Sin[a + b*x],x]

[Out]

-((2^(-1 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hyperge
ometric2F1[-n, (b - d*n)/(2*d), (2 + b/d - n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^n*(b -
d*n))) - (2^(-1 - n)*E^((-I)*(a + c*n) - I*(b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))
^n*Hypergeometric2F1[-n, -1/2*(b + d*n)/d, 1 - (b + d*n)/(2*d), -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I
)*d*x))^n*(b + d*n))

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

Rule 2285

Int[((a_) + (b_.)*(F_)^((e_.)*(v_)))^(p_)*(G_)^((h_.)*(u_)), x_Symbol] :> Int[G^(h*ExpandToSum[u, x])*(a + b*F
^(e*ExpandToSum[v, x]))^p, x] /; FreeQ[{F, G, a, b, e, h, p}, x] && LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v
}, x]

Rule 2323

Int[(u_.)*((a_.)*(F_)^(v_) + (b_.)*(F_)^(w_))^(n_), x_Symbol] :> Dist[(a*F^v + b*F^w)^n/(F^(n*v)*(a + b*F^Expa
ndToSum[w - v, x])^n), Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x], x] /; FreeQ[{F, a, b, n}, x] &&  !
IntegerQ[n] && LinearQ[{v, w}, x]

Rule 4651

Int[Cos[(c_.) + (d_.)*(x_)]^(q_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/2^(p + q), Int[ExpandInte
grand[(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x], x] /; FreeQ
[{a, b, c, d, q}, x] && IGtQ[p, 0] &&  !IntegerQ[q]

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

Mathematica [A] (warning: unable to verify)

Time = 2.47 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.65 \[ \int \cos ^n(c+d x) \sin (a+b x) \, dx=-\frac {2^{-1-n} e^{-i (a-c+(b-d) x)} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{1+n} \left ((b-d n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (2-\frac {b}{d}+n\right ),-\frac {b+d (-2+n)}{2 d},-e^{2 i (c+d x)}\right )+e^{2 i (a+b x)} (b+d n) \operatorname {Hypergeometric2F1}\left (1,\frac {b+d (2+n)}{2 d},\frac {1}{2} \left (2+\frac {b}{d}-n\right ),-e^{2 i (c+d x)}\right )\right )}{(b-d n) (b+d n)} \]

[In]

Integrate[Cos[c + d*x]^n*Sin[a + b*x],x]

[Out]

-((2^(-1 - n)*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + n)*((b - d*n)*Hypergeometric2F1[1, (2 - b/d + n
)/2, -1/2*(b + d*(-2 + n))/d, -E^((2*I)*(c + d*x))] + E^((2*I)*(a + b*x))*(b + d*n)*Hypergeometric2F1[1, (b +
d*(2 + n))/(2*d), (2 + b/d - n)/2, -E^((2*I)*(c + d*x))]))/(E^(I*(a - c + (b - d)*x))*(b - d*n)*(b + d*n)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cos(c + d*x)^n*sin(a + b*x),x)

[Out]

int(cos(c + d*x)^n*sin(a + b*x), x)