3.4 \(\int \frac{(d \sin (e+f x))^n (A+B \sin (e+f x))}{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=202 \[ \frac{(n+1) (A-B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{a d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{a d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (a \sin (e+f x)+a)} \]

[Out]

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

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Rubi [A]  time = 0.223897, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2978, 2748, 2643} \[ \frac{(n+1) (A-B) \cos (e+f x) (d \sin (e+f x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{a d^2 f (n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \cos (e+f x) (d \sin (e+f x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{a d f (n+1) \sqrt{\cos ^2(e+f x)}}+\frac{(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{d f (a \sin (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2978

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])

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

Mathematica [A]  time = 0.866731, size = 157, normalized size = 0.78 \[ \frac{\sin (e+f x) \cos (e+f x) (d \sin (e+f x))^n \left (\frac{(n+1) (A-B) \sin (e+f x) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(e+f x)\right )}{(n+2) \sqrt{\cos ^2(e+f x)}}+\frac{(-A n+B n+B) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(e+f x)\right )}{(n+1) \sqrt{\cos ^2(e+f x)}}+\frac{A-B}{\sin (e+f x)+1}\right )}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\sin \left ( fx+e \right ) \right ) ^{n} \left ( A+B\sin \left ( fx+e \right ) \right ) }{a+a\sin \left ( fx+e \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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