3.69 \(\int \frac{x^2}{\sin ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=141 \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{24 a^3}-\frac{9 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}+\frac{3 x^2 \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{\sqrt{1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}-\frac{x}{3 a^2 \sin ^{-1}(a x)^2}+\frac{x^3}{2 \sin ^{-1}(a x)^2} \]

[Out]

-(x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3) - x/(3*a^2*ArcSin[a*x]^2) + x^3/(2*ArcSin[a*x]^2) - Sqrt[1 - a^2*
x^2]/(3*a^3*ArcSin[a*x]) + (3*x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]) + SinIntegral[ArcSin[a*x]]/(24*a^3) - (
9*SinIntegral[3*ArcSin[a*x]])/(8*a^3)

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Rubi [A]  time = 0.30341, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4633, 4719, 4631, 3299, 4621, 4723} \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{24 a^3}-\frac{9 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}+\frac{3 x^2 \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{\sqrt{1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}-\frac{x}{3 a^2 \sin ^{-1}(a x)^2}+\frac{x^3}{2 \sin ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSin[a*x]^4,x]

[Out]

-(x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3) - x/(3*a^2*ArcSin[a*x]^2) + x^3/(2*ArcSin[a*x]^2) - Sqrt[1 - a^2*
x^2]/(3*a^3*ArcSin[a*x]) + (3*x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]) + SinIntegral[ArcSin[a*x]]/(24*a^3) - (
9*SinIntegral[3*ArcSin[a*x]])/(8*a^3)

Rule 4633

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin
[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n + 1))
/Sqrt[1 - c^2*x^2], x], x] - Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^
2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4719

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)^
(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && GtQ[d, 0]

Rule 4631

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin
[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)
, Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && G
eQ[n, -2] && LtQ[n, -1]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4621

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))
/(b*c*(n + 1)), x] + Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4723

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

Rubi steps

\begin{align*} \int \frac{x^2}{\sin ^{-1}(a x)^4} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}+\frac{2 \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx}{3 a}-a \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{x}{3 a^2 \sin ^{-1}(a x)^2}+\frac{x^3}{2 \sin ^{-1}(a x)^2}-\frac{3}{2} \int \frac{x^2}{\sin ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{\sin ^{-1}(a x)^2} \, dx}{3 a^2}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{x}{3 a^2 \sin ^{-1}(a x)^2}+\frac{x^3}{2 \sin ^{-1}(a x)^2}-\frac{\sqrt{1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac{3 x^2 \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac{3 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 x}+\frac{3 \sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{2 a^3}-\frac{\int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx}{3 a}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{x}{3 a^2 \sin ^{-1}(a x)^2}+\frac{x^3}{2 \sin ^{-1}(a x)^2}-\frac{\sqrt{1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac{3 x^2 \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}-\frac{9 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^3}-\frac{x}{3 a^2 \sin ^{-1}(a x)^2}+\frac{x^3}{2 \sin ^{-1}(a x)^2}-\frac{\sqrt{1-a^2 x^2}}{3 a^3 \sin ^{-1}(a x)}+\frac{3 x^2 \sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)}+\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{24 a^3}-\frac{9 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}\\ \end{align*}

Mathematica [A]  time = 0.25906, size = 102, normalized size = 0.72 \[ \frac{-\frac{8 a^2 x^2 \sqrt{1-a^2 x^2}}{\sin ^{-1}(a x)^3}+\frac{4 a x \left (3 a^2 x^2-2\right )}{\sin ^{-1}(a x)^2}+\frac{4 \sqrt{1-a^2 x^2} \left (9 a^2 x^2-2\right )}{\sin ^{-1}(a x)}+\text{Si}\left (\sin ^{-1}(a x)\right )-27 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{24 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcSin[a*x]^4,x]

[Out]

((-8*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcSin[a*x]^3 + (4*a*x*(-2 + 3*a^2*x^2))/ArcSin[a*x]^2 + (4*Sqrt[1 - a^2*x^2]*
(-2 + 9*a^2*x^2))/ArcSin[a*x] + SinIntegral[ArcSin[a*x]] - 27*SinIntegral[3*ArcSin[a*x]])/(24*a^3)

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Maple [A]  time = 0.028, size = 117, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ( -{\frac{1}{12\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{ax}{24\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}+{\frac{1}{24\,\arcsin \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{24}}+{\frac{\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) }{12\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}}}-{\frac{\sin \left ( 3\,\arcsin \left ( ax \right ) \right ) }{8\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}-{\frac{3\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) }{8\,\arcsin \left ( ax \right ) }}-{\frac{9\,{\it Si} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arcsin(a*x)^4,x)

[Out]

1/a^3*(-1/12/arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)+1/24*a*x/arcsin(a*x)^2+1/24/arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/24*
Si(arcsin(a*x))+1/12/arcsin(a*x)^3*cos(3*arcsin(a*x))-1/8/arcsin(a*x)^2*sin(3*arcsin(a*x))-3/8/arcsin(a*x)*cos
(3*arcsin(a*x))-9/8*Si(3*arcsin(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} \int \frac{{\left (27 \, a^{2} x^{3} - 20 \, x\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{{\left (a^{3} x^{2} - a\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}\,{d x} +{\left (2 \, a^{2} x^{2} -{\left (9 \, a^{2} x^{2} - 2\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}\right )} \sqrt{a x + 1} \sqrt{-a x + 1} -{\left (3 \, a^{3} x^{3} - 2 \, a x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )}{6 \, a^{3} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(6*a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3*integrate(1/6*(27*a^2*x^3 - 20*x)*sqrt(a*x + 1)*sqrt(
-a*x + 1)/((a^3*x^2 - a)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + (2*a^2*x^2 - (9*a^2*x^2 - 2)*arctan
2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) - (3*a^3*x^3 - 2*a*x)*arctan2(a*x, sqrt(a
*x + 1)*sqrt(-a*x + 1)))/(a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{\arcsin \left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^2/arcsin(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{asin}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/asin(a*x)**4,x)

[Out]

Integral(x**2/asin(a*x)**4, x)

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Giac [A]  time = 1.32898, size = 200, normalized size = 1.42 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )^{2}} - \frac{9 \, \operatorname{Si}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac{\operatorname{Si}\left (\arcsin \left (a x\right )\right )}{24 \, a^{3}} - \frac{3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )} + \frac{x}{6 \, a^{2} \arcsin \left (a x\right )^{2}} + \frac{7 \, \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{3} \arcsin \left (a x\right )} + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^4,x, algorithm="giac")

[Out]

1/2*(a^2*x^2 - 1)*x/(a^2*arcsin(a*x)^2) - 9/8*sin_integral(3*arcsin(a*x))/a^3 + 1/24*sin_integral(arcsin(a*x))
/a^3 - 3/2*(-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)) + 1/6*x/(a^2*arcsin(a*x)^2) + 7/6*sqrt(-a^2*x^2 + 1)/(a^3*ar
csin(a*x)) + 1/3*(-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)^3) - 1/3*sqrt(-a^2*x^2 + 1)/(a^3*arcsin(a*x)^3)