Integrand size = 8, antiderivative size = 85 \[ \int x \arcsin (a x)^n \, dx=-\frac {2^{-3-n} (-i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,-2 i \arcsin (a x))}{a^2}-\frac {2^{-3-n} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (1+n,2 i \arcsin (a x))}{a^2} \] Output:
-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)
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int x \arcsin (a x)^n \, dx=-\frac {2^{-3-n} \arcsin (a x)^n \left (\arcsin (a x)^2\right )^{-n} \left ((i \arcsin (a x))^n \Gamma (1+n,-2 i \arcsin (a x))+(-i \arcsin (a x))^n \Gamma (1+n,2 i \arcsin (a x))\right )}{a^2} \] Input:
Integrate[x*ArcSin[a*x]^n,x]
Output:
-((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*(ArcSi n[a*x]^2)^n))
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5146, 4906, 27, 3042, 3789, 2612}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \arcsin (a x)^n \, dx\) |
\(\Big \downarrow \) 5146 |
\(\displaystyle \frac {\int a x \sqrt {1-a^2 x^2} \arcsin (a x)^nd\arcsin (a x)}{a^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {\int \frac {1}{2} \arcsin (a x)^n \sin (2 \arcsin (a x))d\arcsin (a x)}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \arcsin (a x)^n \sin (2 \arcsin (a x))d\arcsin (a x)}{2 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \arcsin (a x)^n \sin (2 \arcsin (a x))d\arcsin (a x)}{2 a^2}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \frac {\frac {1}{2} i \int e^{-2 i \arcsin (a x)} \arcsin (a x)^nd\arcsin (a x)-\frac {1}{2} i \int e^{2 i \arcsin (a x)} \arcsin (a x)^nd\arcsin (a x)}{2 a^2}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle \frac {-2^{-n-2} \arcsin (a x)^n (-i \arcsin (a x))^{-n} \Gamma (n+1,-2 i \arcsin (a x))-2^{-n-2} (i \arcsin (a x))^{-n} \arcsin (a x)^n \Gamma (n+1,2 i \arcsin (a x))}{2 a^2}\) |
Input:
Int[x*ArcSin[a*x]^n,x]
Output:
(-((2^(-2 - n)*ArcSin[a*x]^n*Gamma[1 + n, (-2*I)*ArcSin[a*x]])/((-I)*ArcSi n[a*x])^n) - (2^(-2 - n)*ArcSin[a*x]^n*Gamma[1 + n, (2*I)*ArcSin[a*x]])/(I *ArcSin[a*x])^n)/(2*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.62
method | result | size |
default | \(\frac {\sqrt {\pi }\, \left (\frac {2 \arcsin \left (a x \right )^{1+n} \sin \left (2 \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arcsin \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arcsin \left (a x \right )\right ) \sin \left (2 \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-\sin \left (2 \arcsin \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arcsin \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arcsin \left (a x \right )}}\right )}{4 a^{2}}\) | \(138\) |
Input:
int(x*arcsin(a*x)^n,x,method=_RETURNVERBOSE)
Output:
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)))
\[ \int x \arcsin (a x)^n \, dx=\int { x \arcsin \left (a x\right )^{n} \,d x } \] Input:
integrate(x*arcsin(a*x)^n,x, algorithm="fricas")
Output:
integral(x*arcsin(a*x)^n, x)
\[ \int x \arcsin (a x)^n \, dx=\int x \operatorname {asin}^{n}{\left (a x \right )}\, dx \] Input:
integrate(x*asin(a*x)**n,x)
Output:
Integral(x*asin(a*x)**n, x)
Exception generated. \[ \int x \arcsin (a x)^n \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*arcsin(a*x)^n,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int x \arcsin (a x)^n \, dx=\int { x \arcsin \left (a x\right )^{n} \,d x } \] Input:
integrate(x*arcsin(a*x)^n,x, algorithm="giac")
Output:
integrate(x*arcsin(a*x)^n, x)
Timed out. \[ \int x \arcsin (a x)^n \, dx=\int x\,{\mathrm {asin}\left (a\,x\right )}^n \,d x \] Input:
int(x*asin(a*x)^n,x)
Output:
int(x*asin(a*x)^n, x)
\[ \int x \arcsin (a x)^n \, dx=\int \mathit {asin} \left (a x \right )^{n} x d x \] Input:
int(x*asin(a*x)^n,x)
Output:
int(asin(a*x)**n*x,x)