3.5.98 \(\int \frac {x^3 \text {ArcSin}(a x)^n}{\sqrt {1-a^2 x^2}} \, dx\) [498]

Optimal. Leaf size=163 \[ -\frac {3 (-i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,-i \text {ArcSin}(a x))}{8 a^4}-\frac {3 (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-1-n} (-i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,-3 i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-1-n} (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(1+n,3 i \text {ArcSin}(a x))}{8 a^4} \]

[Out]

-3/8*arcsin(a*x)^n*GAMMA(1+n,-I*arcsin(a*x))/a^4/((-I*arcsin(a*x))^n)-3/8*arcsin(a*x)^n*GAMMA(1+n,I*arcsin(a*x
))/a^4/((I*arcsin(a*x))^n)+1/8*3^(-1-n)*arcsin(a*x)^n*GAMMA(1+n,-3*I*arcsin(a*x))/a^4/((-I*arcsin(a*x))^n)+1/8
*3^(-1-n)*arcsin(a*x)^n*GAMMA(1+n,3*I*arcsin(a*x))/a^4/((I*arcsin(a*x))^n)

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Rubi [A]
time = 0.17, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4809, 3393, 3389, 2212} \begin {gather*} -\frac {3 \text {ArcSin}(a x)^n (-i \text {ArcSin}(a x))^{-n} \text {Gamma}(n+1,-i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-n-1} \text {ArcSin}(a x)^n (-i \text {ArcSin}(a x))^{-n} \text {Gamma}(n+1,-3 i \text {ArcSin}(a x))}{8 a^4}-\frac {3 (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(n+1,i \text {ArcSin}(a x))}{8 a^4}+\frac {3^{-n-1} (i \text {ArcSin}(a x))^{-n} \text {ArcSin}(a x)^n \text {Gamma}(n+1,3 i \text {ArcSin}(a x))}{8 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2],x]

[Out]

(-3*ArcSin[a*x]^n*Gamma[1 + n, (-I)*ArcSin[a*x]])/(8*a^4*((-I)*ArcSin[a*x])^n) - (3*ArcSin[a*x]^n*Gamma[1 + n,
I*ArcSin[a*x]])/(8*a^4*(I*ArcSin[a*x])^n) + (3^(-1 - n)*ArcSin[a*x]^n*Gamma[1 + n, (-3*I)*ArcSin[a*x]])/(8*a^
4*((-I)*ArcSin[a*x])^n) + (3^(-1 - n)*ArcSin[a*x]^n*Gamma[1 + n, (3*I)*ArcSin[a*x]])/(8*a^4*(I*ArcSin[a*x])^n)

Rule 2212

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

Rule 3389

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

Rule 3393

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 4809

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

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 153, normalized size = 0.94 \begin {gather*} -\frac {3^{-1-n} \text {ArcSin}(a x)^n \left (\text {ArcSin}(a x)^2\right )^{-2 n} \left (3^{2+n} (i \text {ArcSin}(a x))^n \left (\text {ArcSin}(a x)^2\right )^n \text {Gamma}(1+n,-i \text {ArcSin}(a x))+(-i \text {ArcSin}(a x))^n \left (3^{2+n} \left (\text {ArcSin}(a x)^2\right )^n \text {Gamma}(1+n,i \text {ArcSin}(a x))-(i \text {ArcSin}(a x))^{2 n} \text {Gamma}(1+n,-3 i \text {ArcSin}(a x))-\left (\text {ArcSin}(a x)^2\right )^n \text {Gamma}(1+n,3 i \text {ArcSin}(a x))\right )\right )}{8 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2],x]

[Out]

-1/8*(3^(-1 - n)*ArcSin[a*x]^n*(3^(2 + n)*(I*ArcSin[a*x])^n*(ArcSin[a*x]^2)^n*Gamma[1 + n, (-I)*ArcSin[a*x]] +
((-I)*ArcSin[a*x])^n*(3^(2 + n)*(ArcSin[a*x]^2)^n*Gamma[1 + n, I*ArcSin[a*x]] - (I*ArcSin[a*x])^(2*n)*Gamma[1
+ n, (-3*I)*ArcSin[a*x]] - (ArcSin[a*x]^2)^n*Gamma[1 + n, (3*I)*ArcSin[a*x]])))/(a^4*(ArcSin[a*x]^2)^(2*n))

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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \arcsin \left (a x \right )^{n}}{\sqrt {-a^{2} x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsin(a*x)^n/(-a^2*x^2+1)^(1/2),x)

[Out]

int(x^3*arcsin(a*x)^n/(-a^2*x^2+1)^(1/2),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^n/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^n/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^3*arcsin(a*x)^n/(a^2*x^2 - 1), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \operatorname {asin}^{n}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*asin(a*x)**n/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**3*asin(a*x)**n/sqrt(-(a*x - 1)*(a*x + 1)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^n/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,{\mathrm {asin}\left (a\,x\right )}^n}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*asin(a*x)^n)/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x^3*asin(a*x)^n)/(1 - a^2*x^2)^(1/2), x)

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