3.132 \(\int x \sin ^{-1}(a x)^n \, dx\)

Optimal. Leaf size=85 \[ -\frac{2^{-n-3} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \sin ^{-1}(a x)\right )}{a^2}-\frac{2^{-n-3} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \sin ^{-1}(a x)\right )}{a^2} \]

[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)

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Rubi [A]  time = 0.0819761, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4635, 4406, 12, 3308, 2181} \[ -\frac{2^{-n-3} \sin ^{-1}(a x)^n \left (-i \sin ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \sin ^{-1}(a x)\right )}{a^2}-\frac{2^{-n-3} \left (i \sin ^{-1}(a x)\right )^{-n} \sin ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \sin ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[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 4635

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

Rule 4406

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 12

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

Rule 3308

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

Mathematica [A]  time = 0.019394, size = 75, normalized size = 0.88 \[ -\frac{2^{-n-3} \sin ^{-1}(a x)^n \left (\sin ^{-1}(a x)^2\right )^{-n} \left (\left (-i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,2 i \sin ^{-1}(a x)\right )+\left (i \sin ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-2 i \sin ^{-1}(a x)\right )\right )}{a^2} \]

Antiderivative was successfully verified.

[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))

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Maple [C]  time = 0.117, size = 138, normalized size = 1.6 \begin{align*}{\frac{\sqrt{\pi }}{4\,{a}^{2}} \left ( 2\,{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{1+n}\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }}-{\frac{\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }{2}^{{\frac{1}{2}}-n}\sqrt{\arcsin \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{3}{2}},2\,\arcsin \left ( ax \right ) \right ) }-3\,{\frac{{2}^{-3/2-n} \left ( 4/3+2/3\,n \right ) \left ( 2\,\arcsin \left ( ax \right ) \cos \left ( 2\,\arcsin \left ( ax \right ) \right ) -\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) \right ){\it LommelS1} \left ( n+1/2,1/2,2\,\arcsin \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) \sqrt{\arcsin \left ( ax \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \arcsin \left (a x\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*arcsin(a*x)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{asin}^{n}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*asin(a*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*arcsin(a*x)^n, x)