3.223 \(\int \cos ^n(c+d x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=568 \[ \frac{2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (\frac{3 b}{d}-n\right ),-n,\frac{1}{2} \left (\frac{3 b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac{3\ 2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,\frac{b-d n}{2 d},\frac{1}{2} \left (\frac{b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{3\ 2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-\frac{b+d n}{2 d},1-\frac{b+d n}{2 d},-e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac{2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-\frac{3 b+d n}{2 d},\frac{1}{2} \left (-\frac{3 b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n} \]

[Out]

(2^(-3 - n)*E^(I*(3*a - c*n) + I*(3*b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hyper
geometric2F1[((3*b)/d - n)/2, -n, (2 + (3*b)/d - n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^n
*(3*b - d*n)) - (3*2^(-3 - 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, (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)) - (3*2^(-3 - 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, -(b + d*n)/(2*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)) + (2^(-3 - n)*E^((-I)*(3*a + c*n) - I*(3*b + d*n)*x + I*n*(c + d*x))*(E^
((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -(3*b + d*n)/(2*d), (2 - (3*b)/d - n)/2, -E^((2*I)
*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^n*(3*b + d*n))

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Rubi [A]  time = 1.18036, antiderivative size = 568, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {4555, 2285, 2253, 2251} \[ \frac{2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac{1}{2} \left (\frac{3 b}{d}-n\right ),-n;\frac{1}{2} \left (\frac{3 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac{3\ 2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,\frac{b-d n}{2 d};\frac{1}{2} \left (\frac{b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{3\ 2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,-\frac{b+d n}{2 d};1-\frac{b+d n}{2 d};-e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac{2^{-n-3} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,-\frac{3 b+d n}{2 d};\frac{1}{2} \left (-\frac{3 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^n*Sin[a + b*x]^3,x]

[Out]

(2^(-3 - n)*E^(I*(3*a - c*n) + I*(3*b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hyper
geometric2F1[((3*b)/d - n)/2, -n, (2 + (3*b)/d - n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^n
*(3*b - d*n)) - (3*2^(-3 - 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, (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)) - (3*2^(-3 - 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, -(b + d*n)/(2*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)) + (2^(-3 - n)*E^((-I)*(3*a + c*n) - I*(3*b + d*n)*x + I*n*(c + d*x))*(E^
((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -(3*b + d*n)/(2*d), (2 - (3*b)/d - n)/2, -E^((2*I)
*(c + d*x))])/((1 + E^((2*I)*c + (2*I)*d*x))^n*(3*b + d*n))

Rule 4555

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]

Rule 2285

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 2253

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 2251

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

Rubi steps

\begin{align*} \int \cos ^n(c+d x) \sin ^3(a+b x) \, dx &=2^{-3-n} \int \left (3 i e^{-i a-i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n-3 i e^{i a+i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n-i e^{-3 i a-3 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n+i e^{3 i a+3 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n\right ) \, dx\\ &=-\left (\left (i 2^{-3-n}\right ) \int e^{-3 i a-3 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx\right )+\left (i 2^{-3-n}\right ) \int e^{3 i a+3 i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx+\left (3 i 2^{-3-n}\right ) \int e^{-i a-i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx-\left (3 i 2^{-3-n}\right ) \int e^{i a+i b x} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \, dx\\ &=-\left (\left (i 2^{-3-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^{-3 i a-3 i b x-i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^n \, dx\right )+\left (i 2^{-3-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^{3 i a+3 i b x-i n (c+d x)} \left (1+e^{2 i c+2 i d x}\right )^n \, dx+\left (3 i 2^{-3-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 (3 i 2^{-3-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^{-3-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 (3 a-c n)+i (3 b-d n) x} \left (1+e^{2 i c+2 i d x}\right )^n \, dx-\left (i 2^{-3-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 (3 a+c n)-i (3 b+d n) x} \left (1+e^{2 i c+2 i d x}\right )^n \, dx-\left (3 i 2^{-3-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+\left (3 i 2^{-3-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^{-3-n} \exp (i (3 a-c n)+i (3 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 \, _2F_1\left (\frac{1}{2} \left (\frac{3 b}{d}-n\right ),-n;\frac{1}{2} \left (2+\frac{3 b}{d}-n\right );-e^{2 i (c+d x)}\right )}{3 b-d n}-\frac{3\ 2^{-3-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 \, _2F_1\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{3\ 2^{-3-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 \, _2F_1\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}+\frac{2^{-3-n} \exp (-i (3 a+c n)-i (3 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 \, _2F_1\left (-n,-\frac{3 b+d n}{2 d};\frac{1}{2} \left (2-\frac{3 b}{d}-n\right );-e^{2 i (c+d x)}\right )}{3 b+d n}\\ \end{align*}

Mathematica [A]  time = 24.6439, size = 329, normalized size = 0.58 \[ 2^{-n-3} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{n+1} e^{i (-3 a+c+d (n+1) x)} \left (-\frac{3 e^{2 i a-i x (b+d n)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (-\frac{b}{d}+n+2\right ),-\frac{b+d (n-2)}{2 d},-e^{2 i (c+d x)}\right )}{b+d n}+e^{i (4 a+b x-d n x)} \left (\frac{e^{2 i (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (\frac{3 b}{d}+n+2\right ),\frac{3 b}{2 d}-\frac{n}{2}+1,-e^{2 i (c+d x)}\right )}{3 b-d n}-\frac{3 \text{Hypergeometric2F1}\left (1,\frac{b+d (n+2)}{2 d},\frac{1}{2} \left (\frac{b}{d}-n+2\right ),-e^{2 i (c+d x)}\right )}{b-d n}\right )+\frac{e^{-i x (3 b+d n)} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (-\frac{3 b}{d}+n+2\right ),-\frac{3 b}{2 d}-\frac{n}{2}+1,-e^{2 i (c+d x)}\right )}{3 b+d n}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^n*Sin[a + b*x]^3,x]

[Out]

2^(-3 - n)*E^(I*(-3*a + c + d*(1 + n)*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + n)*(Hypergeometric2
F1[1, (2 - (3*b)/d + n)/2, 1 - (3*b)/(2*d) - n/2, -E^((2*I)*(c + d*x))]/(E^(I*(3*b + d*n)*x)*(3*b + d*n)) - (3
*E^((2*I)*a - I*(b + d*n)*x)*Hypergeometric2F1[1, (2 - b/d + n)/2, -(b + d*(-2 + n))/(2*d), -E^((2*I)*(c + d*x
))])/(b + d*n) + E^(I*(4*a + b*x - d*n*x))*((E^((2*I)*(a + b*x))*Hypergeometric2F1[1, (2 + (3*b)/d + n)/2, 1 +
(3*b)/(2*d) - n/2, -E^((2*I)*(c + d*x))])/(3*b - d*n) - (3*Hypergeometric2F1[1, (b + d*(2 + n))/(2*d), (2 + b
/d - n)/2, -E^((2*I)*(c + d*x))])/(b - d*n)))

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Maple [F]  time = 0.762, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{n} \left ( \sin \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^n*sin(b*x+a)^3,x)

[Out]

int(cos(d*x+c)^n*sin(b*x+a)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^n*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^n*sin(b*x + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \cos \left (d x + c\right )^{n} \sin \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^n*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(-(cos(b*x + a)^2 - 1)*cos(d*x + c)^n*sin(b*x + a), 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)**n*sin(b*x+a)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (d x + c\right )^{n} \sin \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^n*sin(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^n*sin(b*x + a)^3, x)