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

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

[Out]

5/16*Ci(2*arcsin(a*x))/a^6-1/2*Ci(4*arcsin(a*x))/a^6+3/16*Ci(6*arcsin(a*x))/a^6-x^5*(-a^2*x^2+1)^(1/2)/a/arcsi
n(a*x)

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Rubi [A]  time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [A]  time = 0.04, size = 78, normalized size = 1.10 \[ -\frac {-10 \sin ^{-1}(a x) \text {Ci}\left (2 \sin ^{-1}(a x)\right )+16 \sin ^{-1}(a x) \text {Ci}\left (4 \sin ^{-1}(a x)\right )-6 \sin ^{-1}(a x) \text {Ci}\left (6 \sin ^{-1}(a x)\right )+5 \sin \left (2 \sin ^{-1}(a x)\right )-4 \sin \left (4 \sin ^{-1}(a x)\right )+\sin \left (6 \sin ^{-1}(a x)\right )}{32 a^6 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(-10*ArcSin[a*x]*CosIntegral[2*ArcSin[a*x]] + 16*ArcSin[a*x]*CosIntegral[4*ArcSin[a*x]] - 6*ArcSin[a*x]*
CosIntegral[6*ArcSin[a*x]] + 5*Sin[2*ArcSin[a*x]] - 4*Sin[4*ArcSin[a*x]] + Sin[6*ArcSin[a*x]])/(a^6*ArcSin[a*x
])

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{5}}{\arcsin \left (a x\right )^{2}}, x\right ) \]

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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giac [A]  time = 0.14, size = 120, normalized size = 1.69 \[ -\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x}{a^{5} \arcsin \left (a x\right )} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{5} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{5} \arcsin \left (a x\right )} + \frac {3 \, \operatorname {Ci}\left (6 \, \arcsin \left (a x\right )\right )}{16 \, a^{6}} - \frac {\operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{2 \, a^{6}} + \frac {5 \, \operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{16 \, a^{6}} \]

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

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maple [A]  time = 0.06, size = 78, normalized size = 1.10 \[ \frac {-\frac {5 \sin \left (2 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {5 \Ci \left (2 \arcsin \left (a x \right )\right )}{16}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {\Ci \left (4 \arcsin \left (a x \right )\right )}{2}-\frac {\sin \left (6 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {3 \Ci \left (6 \arcsin \left (a x \right )\right )}{16}}{a^{6}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^5/asin(a*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

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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