\(\int x^3 \arcsin (a x)^n \, dx\) [130]

Optimal result

Integrand size = 10, antiderivative size = 167 \[ \int x^3 \arcsin (a x)^n \, dx=-\frac {2^{-4-n} (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-2 i \arcsin (a x))}{a^4}-\frac {2^{-4-n} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,2 i \arcsin (a x))}{a^4}+\frac {2^{-2 (3+n)} (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-4 i \arcsin (a x))}{a^4}+\frac {2^{-2 (3+n)} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,4 i \arcsin (a x))}{a^4} \]

[Out]

-2^(-4-n)*arcsin(a*x)^n*GAMMA(1+n,-2*I*arcsin(a*x))/a^4/((-I*arcsin(a*x))^n)-2^(-4-n)*arcsin(a*x)^n*GAMMA(1+n,
2*I*arcsin(a*x))/a^4/((I*arcsin(a*x))^n)+arcsin(a*x)^n*GAMMA(1+n,-4*I*arcsin(a*x))/(2^(6+2*n))/a^4/((-I*arcsin
(a*x))^n)+arcsin(a*x)^n*GAMMA(1+n,4*I*arcsin(a*x))/(2^(6+2*n))/a^4/((I*arcsin(a*x))^n)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4731, 4491, 3389, 2212} \[ \int x^3 \arcsin (a x)^n \, dx=-\frac {2^{-n-4} \arcsin (a x)^n (-i \arcsin (a x))^{-n} \Gamma (n+1,-2 i \arcsin (a x))}{a^4}+\frac {2^{-2 (n+3)} \arcsin (a x)^n (-i \arcsin (a x))^{-n} \Gamma (n+1,-4 i \arcsin (a x))}{a^4}-\frac {2^{-n-4} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (n+1,2 i \arcsin (a x))}{a^4}+\frac {2^{-2 (n+3)} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (n+1,4 i \arcsin (a x))}{a^4} \]

[In]

Int[x^3*ArcSin[a*x]^n,x]

[Out]

-((2^(-4 - n)*ArcSin[a*x]^n*Gamma[1 + n, (-2*I)*ArcSin[a*x]])/(a^4*((-I)*ArcSin[a*x])^n)) - (2^(-4 - n)*ArcSin
[a*x]^n*Gamma[1 + n, (2*I)*ArcSin[a*x]])/(a^4*(I*ArcSin[a*x])^n) + (ArcSin[a*x]^n*Gamma[1 + n, (-4*I)*ArcSin[a
*x]])/(2^(2*(3 + n))*a^4*((-I)*ArcSin[a*x])^n) + (ArcSin[a*x]^n*Gamma[1 + n, (4*I)*ArcSin[a*x]])/(2^(2*(3 + n)
)*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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4731

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \cos (x) \sin ^3(x) \, dx,x,\arcsin (a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{4} x^n \sin (2 x)-\frac {1}{8} x^n \sin (4 x)\right ) \, dx,x,\arcsin (a x)\right )}{a^4} \\ & = -\frac {\text {Subst}\left (\int x^n \sin (4 x) \, dx,x,\arcsin (a x)\right )}{8 a^4}+\frac {\text {Subst}\left (\int x^n \sin (2 x) \, dx,x,\arcsin (a x)\right )}{4 a^4} \\ & = -\frac {i \text {Subst}\left (\int e^{-4 i x} x^n \, dx,x,\arcsin (a x)\right )}{16 a^4}+\frac {i \text {Subst}\left (\int e^{4 i x} x^n \, dx,x,\arcsin (a x)\right )}{16 a^4}+\frac {i \text {Subst}\left (\int e^{-2 i x} x^n \, dx,x,\arcsin (a x)\right )}{8 a^4}-\frac {i \text {Subst}\left (\int e^{2 i x} x^n \, dx,x,\arcsin (a x)\right )}{8 a^4} \\ & = -\frac {2^{-4-n} (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-2 i \arcsin (a x))}{a^4}-\frac {2^{-4-n} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,2 i \arcsin (a x))}{a^4}+\frac {4^{-3-n} (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-4 i \arcsin (a x))}{a^4}+\frac {4^{-3-n} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,4 i \arcsin (a x))}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int x^3 \arcsin (a x)^n \, dx=\frac {4^{-3-n} \arcsin (a x)^n \left (\arcsin (a x)^2\right )^{-n} \left (-2^{2+n} (i \arcsin (a x))^n \Gamma (1+n,-2 i \arcsin (a x))-2^{2+n} (-i \arcsin (a x))^n \Gamma (1+n,2 i \arcsin (a x))+(i \arcsin (a x))^n \Gamma (1+n,-4 i \arcsin (a x))+(-i \arcsin (a x))^n \Gamma (1+n,4 i \arcsin (a x))\right )}{a^4} \]

[In]

Integrate[x^3*ArcSin[a*x]^n,x]

[Out]

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

Maple [F]

\[\int x^{3} \arcsin \left (a x \right )^{n}d x\]

[In]

int(x^3*arcsin(a*x)^n,x)

[Out]

int(x^3*arcsin(a*x)^n,x)

Fricas [F]

\[ \int x^3 \arcsin (a x)^n \, dx=\int { x^{3} \arcsin \left (a x\right )^{n} \,d x } \]

[In]

integrate(x^3*arcsin(a*x)^n,x, algorithm="fricas")

[Out]

integral(x^3*arcsin(a*x)^n, x)

Sympy [F]

\[ \int x^3 \arcsin (a x)^n \, dx=\int x^{3} \operatorname {asin}^{n}{\left (a x \right )}\, dx \]

[In]

integrate(x**3*asin(a*x)**n,x)

[Out]

Integral(x**3*asin(a*x)**n, x)

Maxima [F(-2)]

Exception generated. \[ \int x^3 \arcsin (a x)^n \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^3*arcsin(a*x)^n,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int x^3 \arcsin (a x)^n \, dx=\int { x^{3} \arcsin \left (a x\right )^{n} \,d x } \]

[In]

integrate(x^3*arcsin(a*x)^n,x, algorithm="giac")

[Out]

integrate(x^3*arcsin(a*x)^n, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \arcsin (a x)^n \, dx=\int x^3\,{\mathrm {asin}\left (a\,x\right )}^n \,d x \]

[In]

int(x^3*asin(a*x)^n,x)

[Out]

int(x^3*asin(a*x)^n, x)