3.191 \(\int \sin (a+b x) \sin ^n(c+d x) \, dx\)

Optimal. Leaf size=293 \[ -\frac{2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,\frac{b-d n}{2 d},\frac{1}{2} \left (\frac{b}{d}-n+2\right ),e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \text{Hypergeometric2F1}\left (-n,-\frac{b+d n}{2 d},1-\frac{b+d n}{2 d},e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n} \]

[Out]

-((2^(-1 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hyperg
eometric2F1[-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))*(I/E^(I*(c + d*x)) - I*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))

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Rubi [A]  time = 0.84838, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4553, 2285, 2253, 2252, 2251} \[ -\frac{2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,\frac{b-d n}{2 d};\frac{1}{2} \left (\frac{b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (-n,-\frac{b+d n}{2 d};1-\frac{b+d n}{2 d};e^{2 i (c+d x)}\right ) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]*Sin[c + d*x]^n,x]

[Out]

-((2^(-1 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hyperg
eometric2F1[-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))*(I/E^(I*(c + d*x)) - I*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))

Rule 4553

Int[Sin[(a_.) + (b_.)*(x_)]^(p_.)*Sin[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Dist[1/2^(p + q), Int[ExpandInte
grand[(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^q, (I/E^(I*(a + b*x)) - I*E^(I*(a + b*x)))^p, x], x], x] /; Free
Q[{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 2252

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

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

Mathematica [A]  time = 0.857461, size = 209, normalized size = 0.71 \[ \frac{2^{-n-1} e^{-i x (b+d)} \left (-1+e^{2 i (c+d x)}\right ) \left (-i e^{-i (c+d x)} \left (-1+e^{2 i (c+d x)}\right )\right )^n \left (e^{i d x} (\cos (a)-i \sin (a)) (b-d n) \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (-\frac{b}{d}+n+2\right ),-\frac{b+d (n-2)}{2 d},e^{2 i (c+d x)}\right )+(\cos (a)+i \sin (a)) (b+d n) e^{i x (2 b+d)} \text{Hypergeometric2F1}\left (1,\frac{b+d (n+2)}{2 d},\frac{1}{2} \left (\frac{b}{d}-n+2\right ),e^{2 i (c+d x)}\right )\right )}{(b-d n) (b+d n)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.304, size = 0, normalized size = 0. \begin{align*} \int \sin \left ( bx+a \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \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(sin(b*x+a)*sin(d*x+c)^n,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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