3.3 \(\int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx\)

Optimal. Leaf size=191 \[ \frac{a (A+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{a (A (n+2)+B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1) (n+2) \sqrt{\cos ^2(e+f x)}}-\frac{a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2)} \]

[Out]

-((a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n))) + (a*(B*(1 + n) + A*(2 + n))*Cos[e + f*x]*Hyperge
ometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*Sqrt[Cos[
e + f*x]^2]) + (a*(A + B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e +
f*x])^(2 + n))/(d^2*f*(2 + n)*Sqrt[Cos[e + f*x]^2])

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Rubi [A]  time = 0.217322, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {2968, 3023, 2748, 2643} \[ \frac{a (A+B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{a (A (n+2)+B (n+1)) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{d f (n+1) (n+2) \sqrt{\cos ^2(e+f x)}}-\frac{a B \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]),x]

[Out]

-((a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n))) + (a*(B*(1 + n) + A*(2 + n))*Cos[e + f*x]*Hyperge
ometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*Sqrt[Cos[
e + f*x]^2]) + (a*(A + B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e +
f*x])^(2 + n))/(d^2*f*(2 + n)*Sqrt[Cos[e + f*x]^2])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (d \sin (e+f x))^n (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx &=\int (d \sin (e+f x))^n \left (a A+(a A+a B) \sin (e+f x)+a B \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac{\int (d \sin (e+f x))^n (a d (B (1+n)+A (2+n))+a (A+B) d (2+n) \sin (e+f x)) \, dx}{d (2+n)}\\ &=-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac{(a (A+B)) \int (d \sin (e+f x))^{1+n} \, dx}{d}+\frac{(a (B (1+n)+A (2+n))) \int (d \sin (e+f x))^n \, dx}{2+n}\\ &=-\frac{a B \cos (e+f x) (d \sin (e+f x))^{1+n}}{d f (2+n)}+\frac{a (B (1+n)+A (2+n)) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+n}}{d f (1+n) (2+n) \sqrt{\cos ^2(e+f x)}}+\frac{a (A+B) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{2+n}{2};\frac{4+n}{2};\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+n}}{d^2 f (2+n) \sqrt{\cos ^2(e+f x)}}\\ \end{align*}

Mathematica [C]  time = 3.77841, size = 392, normalized size = 2.05 \[ -\frac{a 2^{-n-2} e^{i f n x} \left (1-e^{2 i (e+f x)}\right )^{-n} \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^n (\sin (e+f x)+1) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \left (\frac{2 (A+B) e^{-i (e+f (n+1) x)} \, _2F_1\left (\frac{1}{2} (-n-1),-n;\frac{1-n}{2};e^{2 i (e+f x)}\right )}{n+1}-\frac{2 (A+B) e^{i (e-f (n-1) x)} \, _2F_1\left (\frac{1-n}{2},-n;\frac{3-n}{2};e^{2 i (e+f x)}\right )}{n-1}+i \left (\frac{e^{-i f n x} \left (B n e^{2 i (e+f x)} \, _2F_1\left (1-\frac{n}{2},-n;2-\frac{n}{2};e^{2 i (e+f x)}\right )-2 (n-2) (2 A+B) \, _2F_1\left (-n,-\frac{n}{2};1-\frac{n}{2};e^{2 i (e+f x)}\right )\right )}{(n-2) n}+\frac{B e^{-i (2 e+f (n+2) x)} \, _2F_1\left (-\frac{n}{2}-1,-n;-\frac{n}{2};e^{2 i (e+f x)}\right )}{n+2}\right )\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]),x]

[Out]

-((2^(-2 - n)*a*E^(I*f*n*x)*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^(I*(e + f*x)))^n*((2*(A + B)*Hypergeometric2F
1[(-1 - n)/2, -n, (1 - n)/2, E^((2*I)*(e + f*x))])/(E^(I*(e + f*(1 + n)*x))*(1 + n)) - (2*(A + B)*E^(I*(e - f*
(-1 + n)*x))*Hypergeometric2F1[(1 - n)/2, -n, (3 - n)/2, E^((2*I)*(e + f*x))])/(-1 + n) + I*((B*Hypergeometric
2F1[-1 - n/2, -n, -n/2, E^((2*I)*(e + f*x))])/(E^(I*(2*e + f*(2 + n)*x))*(2 + n)) + (B*E^((2*I)*(e + f*x))*n*H
ypergeometric2F1[1 - n/2, -n, 2 - n/2, E^((2*I)*(e + f*x))] - 2*(2*A + B)*(-2 + n)*Hypergeometric2F1[-n, -n/2,
1 - n/2, E^((2*I)*(e + f*x))])/(E^(I*f*n*x)*(-2 + n)*n)))*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x]))/((1 - E^((2*
I)*(e + f*x)))^n*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sin[e + f*x]^n))

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Maple [F]  time = 1.806, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) \left ( A+B\sin \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))*(A+B*sin(f*x+e)),x)

[Out]

int((d*sin(f*x+e))^n*(a+a*sin(f*x+e))*(A+B*sin(f*x+e)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral(-(B*a*cos(f*x + e)^2 - (A + B)*a*sin(f*x + e) - (A + B)*a)*(d*sin(f*x + e))^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((d*sin(f*x+e))**n*(a+a*sin(f*x+e))*(A+B*sin(f*x+e)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n, x)