3.62 \(\int \frac {x^2}{\sin ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=82 \[ -\frac {\text {Ci}\left (\sin ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Ci}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac {x}{a^2 \sin ^{-1}(a x)}+\frac {3 x^3}{2 \sin ^{-1}(a x)} \]

[Out]

-x/a^2/arcsin(a*x)+3/2*x^3/arcsin(a*x)-1/8*Ci(arcsin(a*x))/a^3+9/8*Ci(3*arcsin(a*x))/a^3-1/2*x^2*(-a^2*x^2+1)^
(1/2)/a/arcsin(a*x)^2

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Rubi [A]  time = 0.25, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4633, 4719, 4635, 4406, 3302, 4623} \[ -\frac {\text {CosIntegral}\left (\sin ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{8 a^3}-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac {x}{a^2 \sin ^{-1}(a x)}+\frac {3 x^3}{2 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSin[a*x]^3,x]

[Out]

-(x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]^2) - x/(a^2*ArcSin[a*x]) + (3*x^3)/(2*ArcSin[a*x]) - CosIntegral[Arc
Sin[a*x]]/(8*a^3) + (9*CosIntegral[3*ArcSin[a*x]])/(8*a^3)

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 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 4623

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

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 68, normalized size = 0.83 \[ \frac {\frac {4 a x \left (\left (3 a^2 x^2-2\right ) \sin ^{-1}(a x)-a x \sqrt {1-a^2 x^2}\right )}{\sin ^{-1}(a x)^2}-\text {Ci}\left (\sin ^{-1}(a x)\right )+9 \text {Ci}\left (3 \sin ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcSin[a*x]^3,x]

[Out]

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

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arcsin \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^2/arcsin(a*x)^3, x)

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giac [A]  time = 0.21, size = 102, normalized size = 1.24 \[ \frac {3 \, {\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )} + \frac {x}{2 \, a^{2} \arcsin \left (a x\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} \arcsin \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^3,x, algorithm="giac")

[Out]

3/2*(a^2*x^2 - 1)*x/(a^2*arcsin(a*x)) + 1/2*x/(a^2*arcsin(a*x)) + 9/8*cos_integral(3*arcsin(a*x))/a^3 - 1/8*co
s_integral(arcsin(a*x))/a^3 + 1/2*(-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)^2) - 1/2*sqrt(-a^2*x^2 + 1)/(a^3*arcsi
n(a*x)^2)

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maple [A]  time = 0.03, size = 82, normalized size = 1.00 \[ \frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arcsin \left (a x \right )^{2}}+\frac {a x}{8 \arcsin \left (a x \right )}-\frac {\Ci \left (\arcsin \left (a x \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}+\frac {9 \Ci \left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arcsin(a*x)^3,x)

[Out]

1/a^3*(-1/8/arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+1/8*a*x/arcsin(a*x)-1/8*Ci(arcsin(a*x))+1/8/arcsin(a*x)^2*cos(3*a
rcsin(a*x))-3/8/arcsin(a*x)*sin(3*arcsin(a*x))+9/8*Ci(3*arcsin(a*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\sqrt {a x + 1} \sqrt {-a x + 1} a x^{2} + \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} \int \frac {9 \, a^{2} x^{2} - 2}{\arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} - {\left (3 \, a^{2} x^{3} - 2 \, x\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}{2 \, a^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^3,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/asin(a*x)^3,x)

[Out]

int(x^2/asin(a*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asin}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/asin(a*x)**3,x)

[Out]

Integral(x**2/asin(a*x)**3, x)

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