3.219 \(\int \cos ^n(c+d x) \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=386 \[ -\frac{i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (-\frac{2 b}{d}-n\right ),-n,\frac{1}{2} \left (-\frac{2 b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac{i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (\frac{2 b}{d}-n\right ),-n,\frac{1}{2} \left (\frac{2 b}{d}-n+2\right ),-e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac{i 2^{-n-1} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i (c+d x)}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-\frac{n}{2},1-\frac{n}{2},-e^{2 i (c+d x)}\right )}{d n} \]

[Out]

((-I)*2^(-2 - n)*E^((-I)*(2*a + c*n) - I*(2*b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))
^n*Hypergeometric2F1[((-2*b)/d - n)/2, -n, (2 - (2*b)/d - n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I
)*d*x))^n*(2*b + d*n)) + (I*2^(-2 - n)*E^(I*(2*a - c*n) + I*(2*b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x))
+ E^(I*(c + d*x)))^n*Hypergeometric2F1[((2*b)/d - n)/2, -n, (2 + (2*b)/d - n)/2, -E^((2*I)*(c + d*x))])/((1 +
E^((2*I)*c + (2*I)*d*x))^n*(2*b - d*n)) + (I*2^(-1 - n)*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometr
ic2F1[-n, -n/2, 1 - n/2, -E^((2*I)*(c + d*x))])/(d*(1 + E^((2*I)*(c + d*x)))^n*n)

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Rubi [A]  time = 0.700175, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {4555, 2282, 2032, 364, 2285, 2253, 2251} \[ -\frac{i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac{1}{2} \left (-\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (-\frac{2 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac{i 2^{-n-2} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac{1}{2} \left (\frac{2 b}{d}-n\right ),-n;\frac{1}{2} \left (\frac{2 b}{d}-n+2\right );-e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac{i 2^{-n-1} \left (e^{-i (c+d x)}+e^{i (c+d x)}\right )^n \left (1+e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac{n}{2};1-\frac{n}{2};-e^{2 i (c+d x)}\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^n*Sin[a + b*x]^2,x]

[Out]

((-I)*2^(-2 - n)*E^((-I)*(2*a + c*n) - I*(2*b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))
^n*Hypergeometric2F1[((-2*b)/d - n)/2, -n, (2 - (2*b)/d - n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*c + (2*I
)*d*x))^n*(2*b + d*n)) + (I*2^(-2 - n)*E^(I*(2*a - c*n) + I*(2*b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x))
+ E^(I*(c + d*x)))^n*Hypergeometric2F1[((2*b)/d - n)/2, -n, (2 + (2*b)/d - n)/2, -E^((2*I)*(c + d*x))])/((1 +
E^((2*I)*c + (2*I)*d*x))^n*(2*b - d*n)) + (I*2^(-1 - n)*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometr
ic2F1[-n, -n/2, 1 - n/2, -E^((2*I)*(c + d*x))])/(d*(1 + E^((2*I)*(c + d*x)))^n*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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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

Mathematica [A]  time = 1.78569, size = 242, normalized size = 0.63 \[ -\frac{i 2^{-n-2} \left (e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )^{n+1} e^{i (c+d x)-2 i (a+b x)} \left (e^{2 i (a+b x)} (2 b+d n) \left (d n e^{2 i (a+b x)} \text{Hypergeometric2F1}\left (1,\frac{b}{d}+\frac{n}{2}+1,\frac{b}{d}-\frac{n}{2}+1,-e^{2 i (c+d x)}\right )+2 (2 b-d n) \text{Hypergeometric2F1}\left (1,\frac{n+2}{2},1-\frac{n}{2},-e^{2 i (c+d x)}\right )\right )+d n (d n-2 b) \text{Hypergeometric2F1}\left (1,-\frac{b}{d}+\frac{n}{2}+1,-\frac{b}{d}-\frac{n}{2}+1,-e^{2 i (c+d x)}\right )\right )}{d^3 n^3-4 b^2 d n} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^n*Sin[a + b*x]^2,x]

[Out]

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

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Maple [F]  time = 0.751, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{n} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^n*sin(b*x+a)^2,x)

[Out]

int(cos(d*x+c)^n*sin(b*x+a)^2,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 )^{2}\,{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)^2,x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^n*sin(b*x + a)^2, 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}, 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)^2,x, algorithm="fricas")

[Out]

integral(-(cos(b*x + a)^2 - 1)*cos(d*x + c)^n, 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)**2,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 )^{2}\,{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)^2,x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^n*sin(b*x + a)^2, x)