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

Optimal. Leaf size=410 \[ -\frac {i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (-\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (-\frac {2 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac {i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (\frac {2 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac {i 2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};e^{2 i (c+d x)}\right )}{d n} \]

[Out]

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

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Rubi [A]  time = 0.97, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {4553, 2282, 1980, 2032, 365, 364, 2285, 2253, 2252, 2251} \[ -\frac {i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (-\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (-\frac {2 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac {i 2^{-n-2} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i c+2 i d x}\right )^{-n} \, _2F_1\left (\frac {1}{2} \left (\frac {2 b}{d}-n\right ),-n;\frac {1}{2} \left (\frac {2 b}{d}-n+2\right );e^{2 i (c+d x)}\right ) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac {i 2^{-n-1} \left (i e^{-i (c+d x)}-i e^{i (c+d x)}\right )^n \left (1-e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};e^{2 i (c+d x)}\right )}{d n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^2*Sin[c + d*x]^n,x]

[Out]

((-I)*2^(-2 - n)*E^((-I)*(2*a + c*n) - I*(2*b + d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*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))*(I/E^(I*(c + d*x))
- I*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)*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeomet
ric2F1[-n, -n/2, 1 - n/2, E^((2*I)*(c + d*x))])/(d*(1 - E^((2*I)*(c + d*x)))^n*n)

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 365

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

Rule 1980

Int[(u_)^(p_.)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] &&
GeneralizedBinomialQ[u, x] &&  !GeneralizedBinomialMatchQ[u, 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 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])

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 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 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 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 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]

Rubi steps

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

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \[ \int \sin ^2(a+b x) \sin ^n(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (d x + c\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 7.18, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{2}\left (b x +a \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin \left (d x + c\right )^{n} \sin \left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\sin \left (a+b\,x\right )}^2\,{\sin \left (c+d\,x\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^2*sin(c + d*x)^n,x)

[Out]

int(sin(a + b*x)^2*sin(c + d*x)^n, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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