### 3.71 $$\int (-4+3 \sin (c+d x))^n \, dx$$

Optimal. Leaf size=95 $\frac{\sqrt{2} 7^n \cos (c+d x) (4-3 \sin (c+d x))^{-n} (3 \sin (c+d x)-4)^n F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}}$

[Out]

(Sqrt[2]*7^n*AppellF1[1/2, -n, 1/2, 3/2, (3*(1 + Sin[c + d*x]))/7, (1 + Sin[c + d*x])/2]*Cos[c + d*x]*(-4 + 3*
Sin[c + d*x])^n)/(d*(4 - 3*Sin[c + d*x])^n*Sqrt[1 - Sin[c + d*x]])

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Rubi [A]  time = 0.0509589, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {2665, 139, 138} $\frac{\sqrt{2} 7^n \cos (c+d x) (4-3 \sin (c+d x))^{-n} (3 \sin (c+d x)-4)^n F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-4 + 3*Sin[c + d*x])^n,x]

[Out]

(Sqrt[2]*7^n*AppellF1[1/2, -n, 1/2, 3/2, (3*(1 + Sin[c + d*x]))/7, (1 + Sin[c + d*x])/2]*Cos[c + d*x]*(-4 + 3*
Sin[c + d*x])^n)/(d*(4 - 3*Sin[c + d*x])^n*Sqrt[1 - Sin[c + d*x]])

Rule 2665

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt
[1 - Sin[c + d*x]]), Subst[Int[(a + b*x)^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b,
c, d, n}, x] && NeQ[a^2 - b^2, 0] &&  !IntegerQ[2*n]

Rule 139

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*((b*(e + f*x))/(b*e - a*f))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
((b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rubi steps

\begin{align*} \int (-4+3 \sin (c+d x))^n \, dx &=\frac{\cos (c+d x) \operatorname{Subst}\left (\int \frac{(-4+3 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} \sqrt{1+\sin (c+d x)}}\\ &=\frac{\left (\cos (c+d x) (4-3 \sin (c+d x))^{-n} (-4+3 \sin (c+d x))^n\right ) \operatorname{Subst}\left (\int \frac{(4-3 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} \sqrt{1+\sin (c+d x)}}\\ &=\frac{\sqrt{2} 7^n F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (1+\sin (c+d x)),\frac{1}{2} (1+\sin (c+d x))\right ) \cos (c+d x) (4-3 \sin (c+d x))^{-n} (-4+3 \sin (c+d x))^n}{d \sqrt{1-\sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.119194, size = 95, normalized size = 1. $\frac{\sqrt{\sin (c+d x)-1} \sqrt{\sin (c+d x)+1} \sec (c+d x) (3 \sin (c+d x)-4)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right )}{\sqrt{7} d (n+1)}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(-4 + 3*Sin[c + d*x])^n,x]

[Out]

(AppellF1[1 + n, 1/2, 1/2, 2 + n, (4 - 3*Sin[c + d*x])/7, 4 - 3*Sin[c + d*x]]*Sec[c + d*x]*Sqrt[-1 + Sin[c + d
*x]]*Sqrt[1 + Sin[c + d*x]]*(-4 + 3*Sin[c + d*x])^(1 + n))/(Sqrt[7]*d*(1 + n))

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Maple [F]  time = 0.268, size = 0, normalized size = 0. \begin{align*} \int \left ( -4+3\,\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-4+3*sin(d*x+c))^n,x)

[Out]

int((-4+3*sin(d*x+c))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (3 \, \sin \left (d x + c\right ) - 4\right )}^{n}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+3*sin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((3*sin(d*x + c) - 4)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (3 \, \sin \left (d x + c\right ) - 4\right )}^{n}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+3*sin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((3*sin(d*x + c) - 4)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (3 \sin{\left (c + d x \right )} - 4\right )^{n}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+3*sin(d*x+c))**n,x)

[Out]

Integral((3*sin(c + d*x) - 4)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (3 \, \sin \left (d x + c\right ) - 4\right )}^{n}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4+3*sin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((3*sin(d*x + c) - 4)^n, x)