\(\int x \arcsin (a x)^n \, dx\) [132]

Optimal result

Integrand size = 8, antiderivative size = 85 \[ \int x \arcsin (a x)^n \, dx=-\frac {2^{-3-n} (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-2 i \arcsin (a x))}{a^2}-\frac {2^{-3-n} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,2 i \arcsin (a x))}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4731, 4491, 12, 3389, 2212} \[ \int x \arcsin (a x)^n \, dx=-\frac {2^{-n-3} \arcsin (a x)^n (-i \arcsin (a x))^{-n} \Gamma (n+1,-2 i \arcsin (a x))}{a^2}-\frac {2^{-n-3} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (n+1,2 i \arcsin (a x))}{a^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int x \arcsin (a x)^n \, dx=-\frac {2^{-3-n} \arcsin (a x)^n \left (\arcsin (a x)^2\right )^{-n} \left ((i \arcsin (a x))^n \Gamma (1+n,-2 i \arcsin (a x))+(-i \arcsin (a x))^n \Gamma (1+n,2 i \arcsin (a x))\right )}{a^2} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.62

method	result	size
default	\(\frac {\sqrt {\pi }\, \left (\frac {2 \arcsin \left (a x \right )^{1+n} \sin \left (2 \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arcsin \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arcsin \left (a x \right )\right ) \sin \left (2 \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-\sin \left (2 \arcsin \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arcsin \left (a x \right )}}\right )}{4 a^{2}}\)	\(138\)

[In]

int(x*arcsin(a*x)^n,x,method=_RETURNVERBOSE)

[Out]

1/4*Pi^(1/2)/a^2*(2/Pi^(1/2)/(2+n)*arcsin(a*x)^(1+n)*sin(2*arcsin(a*x))-2^(1/2-n)/Pi^(1/2)/(2+n)*arcsin(a*x)^(
1/2)*LommelS1(n+3/2,3/2,2*arcsin(a*x))*sin(2*arcsin(a*x))-3*2^(-3/2-n)/Pi^(1/2)/(2+n)/arcsin(a*x)^(1/2)*(4/3+2
/3*n)*(2*arcsin(a*x)*cos(2*arcsin(a*x))-sin(2*arcsin(a*x)))*LommelS1(n+1/2,1/2,2*arcsin(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x*arcsin(a*x)^n, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*asin(a*x)**n, x)

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x*arcsin(a*x)^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*asin(a*x)^n, x)