3.489 \(\int (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^n \, dx\)

Optimal. Leaf size=466 \[ -\frac{i d 2^{-n-3} e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}-\frac{i d 2^{-2 (n+3)} e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{i d 2^{-n-3} e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{i d 2^{-2 (n+3)} e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt{1-c^2 x^2}} \]

[Out]

(3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*c*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-3 - n)*d*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(c*E^(((2*I)*a)/b)*Sqrt[1
- c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-3 - n)*d*E^(((2*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(c*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) -
(I*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*c*
E^(((4*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*d*E^(((4*I)*a)/b)*Sqrt[d - c^2*d*x^2]
*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*c*Sqrt[1 - c^2*x^2]*((I*(a
+ b*ArcSin[c*x]))/b)^n)

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Rubi [A]  time = 0.406926, antiderivative size = 466, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {4663, 4661, 3312, 3307, 2181} \[ -\frac{i d 2^{-n-3} e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}-\frac{i d 2^{-2 (n+3)} e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{i d 2^{-n-3} e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{i d 2^{-2 (n+3)} e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]

[Out]

(3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*c*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-3 - n)*d*Sqrt
[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(c*E^(((2*I)*a)/b)*Sqrt[1
- c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-3 - n)*d*E^(((2*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcS
in[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(c*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) -
(I*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*c*
E^(((4*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*d*E^(((4*I)*a)/b)*Sqrt[d - c^2*d*x^2]
*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*c*Sqrt[1 - c^2*x^2]*((I*(a
+ b*ArcSin[c*x]))/b)^n)

Rule 4663

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(d^(p - 1/2)*Sqrt[
d + e*x^2])/Sqrt[1 - c^2*x^2], Int[(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0] &&  !(IntegerQ[p] || GtQ[d, 0])

Rule 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cos ^4(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8} (a+b x)^n+\frac{1}{2} (a+b x)^n \cos (2 x)+\frac{1}{8} (a+b x)^n \cos (4 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt{1-c^2 x^2}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cos (4 x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c \sqrt{1-c^2 x^2}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cos (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt{1-c^2 x^2}}\\ &=\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt{1-c^2 x^2}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-4 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c \sqrt{1-c^2 x^2}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{4 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c \sqrt{1-c^2 x^2}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt{1-c^2 x^2}}+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt{1-c^2 x^2}}\\ &=\frac{3 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt{1-c^2 x^2}}-\frac{i 2^{-3-n} d e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{i 2^{-3-n} d e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}-\frac{i 4^{-3-n} d e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}+\frac{i 4^{-3-n} d e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.65971, size = 326, normalized size = 0.7 \[ \frac{d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (i 4^{-n} e^{-\frac{4 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (e^{\frac{8 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )+8 \left (-i 2^{-n} e^{-\frac{2 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 2^{-n} e^{\frac{2 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\frac{4 a+4 b \sin ^{-1}(c x)}{b n+b}\right )-\frac{8 \left (a+b \sin ^{-1}(c x)\right )}{b n+b}\right )}{64 c \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]

[Out]

(d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((-8*(a + b*ArcSin[c*x]))/(b + b*n) + 8*((4*a + 4*b*ArcSin[c*x])/
(b + b*n) - (I*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/
b)^n) + (I*E^(((2*I)*a)/b)*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*((I*(a + b*ArcSin[c*x]))/b)^n)) +
(I*(-(((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b]) + E^(((8*I)*a)/b)*(((-I)*(a
+ b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b]))/(4^n*E^(((4*I)*a)/b)*((a + b*ArcSin[c*x]
)^2/b^2)^n)))/(64*c*Sqrt[d - c^2*d*x^2])

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Maple [F]  time = 0.133, size = 0, normalized size = 0. \begin{align*} \int \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x)

[Out]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x, algorithm="fricas")

[Out]

integral((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^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((-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^n,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^n, x)