3.52 \(\int \frac{x^5}{\sin ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=71 \[ \frac{5 \text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{16 a^6}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{2 a^6}+\frac{3 \text{CosIntegral}\left (6 \sin ^{-1}(a x)\right )}{16 a^6}-\frac{x^5 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

[Out]

-((x^5*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + (5*CosIntegral[2*ArcSin[a*x]])/(16*a^6) - CosIntegral[4*ArcSin[a*
x]]/(2*a^6) + (3*CosIntegral[6*ArcSin[a*x]])/(16*a^6)

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Rubi [A]  time = 0.0634923, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4631, 3302} \[ \frac{5 \text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{16 a^6}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{2 a^6}+\frac{3 \text{CosIntegral}\left (6 \sin ^{-1}(a x)\right )}{16 a^6}-\frac{x^5 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^5/ArcSin[a*x]^2,x]

[Out]

-((x^5*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + (5*CosIntegral[2*ArcSin[a*x]])/(16*a^6) - CosIntegral[4*ArcSin[a*
x]]/(2*a^6) + (3*CosIntegral[6*ArcSin[a*x]])/(16*a^6)

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 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x^5}{\sin ^{-1}(a x)^2} \, dx &=-\frac{x^5 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (\frac{5 \cos (2 x)}{16 x}-\frac{\cos (4 x)}{2 x}+\frac{3 \cos (6 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^6}\\ &=-\frac{x^5 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (6 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^6}+\frac{5 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^6}\\ &=-\frac{x^5 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{5 \text{Ci}\left (2 \sin ^{-1}(a x)\right )}{16 a^6}-\frac{\text{Ci}\left (4 \sin ^{-1}(a x)\right )}{2 a^6}+\frac{3 \text{Ci}\left (6 \sin ^{-1}(a x)\right )}{16 a^6}\\ \end{align*}

Mathematica [A]  time = 0.0445183, size = 78, normalized size = 1.1 \[ -\frac{-10 \sin ^{-1}(a x) \text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )+16 \sin ^{-1}(a x) \text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )-6 \sin ^{-1}(a x) \text{CosIntegral}\left (6 \sin ^{-1}(a x)\right )+5 \sin \left (2 \sin ^{-1}(a x)\right )-4 \sin \left (4 \sin ^{-1}(a x)\right )+\sin \left (6 \sin ^{-1}(a x)\right )}{32 a^6 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/ArcSin[a*x]^2,x]

[Out]

-(-10*ArcSin[a*x]*CosIntegral[2*ArcSin[a*x]] + 16*ArcSin[a*x]*CosIntegral[4*ArcSin[a*x]] - 6*ArcSin[a*x]*CosIn
tegral[6*ArcSin[a*x]] + 5*Sin[2*ArcSin[a*x]] - 4*Sin[4*ArcSin[a*x]] + Sin[6*ArcSin[a*x]])/(32*a^6*ArcSin[a*x])

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Maple [A]  time = 0.033, size = 78, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{6}} \left ( -{\frac{5\,\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{32\,\arcsin \left ( ax \right ) }}+{\frac{5\,{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) }{16}}+{\frac{\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) }{8\,\arcsin \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 4\,\arcsin \left ( ax \right ) \right ) }{2}}-{\frac{\sin \left ( 6\,\arcsin \left ( ax \right ) \right ) }{32\,\arcsin \left ( ax \right ) }}+{\frac{3\,{\it Ci} \left ( 6\,\arcsin \left ( ax \right ) \right ) }{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/arcsin(a*x)^2,x)

[Out]

1/a^6*(-5/32/arcsin(a*x)*sin(2*arcsin(a*x))+5/16*Ci(2*arcsin(a*x))+1/8/arcsin(a*x)*sin(4*arcsin(a*x))-1/2*Ci(4
*arcsin(a*x))-1/32/arcsin(a*x)*sin(6*arcsin(a*x))+3/16*Ci(6*arcsin(a*x)))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5}}{\arcsin \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^5/arcsin(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/asin(a*x)**2,x)

[Out]

Integral(x**5/asin(a*x)**2, x)

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Giac [A]  time = 1.31122, size = 162, normalized size = 2.28 \begin{align*} -\frac{{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} x}{a^{5} \arcsin \left (a x\right )} + \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{a^{5} \arcsin \left (a x\right )} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{5} \arcsin \left (a x\right )} + \frac{3 \, \operatorname{Ci}\left (6 \, \arcsin \left (a x\right )\right )}{16 \, a^{6}} - \frac{\operatorname{Ci}\left (4 \, \arcsin \left (a x\right )\right )}{2 \, a^{6}} + \frac{5 \, \operatorname{Ci}\left (2 \, \arcsin \left (a x\right )\right )}{16 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/arcsin(a*x)^2,x, algorithm="giac")

[Out]

-(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*x/(a^5*arcsin(a*x)) + 2*(-a^2*x^2 + 1)^(3/2)*x/(a^5*arcsin(a*x)) - sqrt(-a
^2*x^2 + 1)*x/(a^5*arcsin(a*x)) + 3/16*cos_integral(6*arcsin(a*x))/a^6 - 1/2*cos_integral(4*arcsin(a*x))/a^6 +
5/16*cos_integral(2*arcsin(a*x))/a^6