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

Optimal. Leaf size=125 \[ -\frac{\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}+\frac{\sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^3}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcSin[a*x]]) + (4*x^3)/Sqrt[ArcSin[a*x
]] - (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin
[a*x]]])/a^3

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Rubi [A]  time = 0.301204, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4633, 4719, 4635, 4406, 3304, 3352, 4623} \[ -\frac{\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}+\frac{\sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^3}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcSin[a*x]]) + (4*x^3)/Sqrt[ArcSin[a*x
]] - (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin
[a*x]]])/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 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 3304

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

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

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]

Rubi steps

\begin{align*} \int \frac{x^2}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{3 a}-(2 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}}-12 \int \frac{x^2}{\sqrt{\sin ^{-1}(a x)}} \, dx+\frac{8 \int \frac{1}{\sqrt{\sin ^{-1}(a x)}} \, dx}{3 a^2}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^3}-\frac{12 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}-\frac{12 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{x}}-\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}-\frac{6 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^3}+\frac{6 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{4 x^3}{\sqrt{\sin ^{-1}(a x)}}-\frac{\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^3}+\frac{\sqrt{6 \pi } C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^3}\\ \end{align*}

Mathematica [C]  time = 0.207792, size = 277, normalized size = 2.22 \[ \frac{\frac{i e^{i \sin ^{-1}(a x)} \left (-2 \sin ^{-1}(a x)+i\right )-2 \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )}{12 \sin ^{-1}(a x)^{3/2}}-\frac{e^{-i \sin ^{-1}(a x)} \left (2 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )-2 i \sin ^{-1}(a x)+1\right )}{12 \sin ^{-1}(a x)^{3/2}}-\frac{i e^{3 i \sin ^{-1}(a x)} \left (-6 \sin ^{-1}(a x)+i\right )-6 \sqrt{3} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )}{12 \sin ^{-1}(a x)^{3/2}}+\frac{e^{-3 i \sin ^{-1}(a x)} \left (6 \sqrt{3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )-6 i \sin ^{-1}(a x)+1\right )}{12 \sin ^{-1}(a x)^{3/2}}}{a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I*E^(I*ArcSin[a*x])*(I - 2*ArcSin[a*x]) - 2*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-I)*ArcSin[a*x]])/(12*ArcSi
n[a*x]^(3/2)) - (1 - (2*I)*ArcSin[a*x] + 2*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, I*ArcSin[a*x]])/
(12*E^(I*ArcSin[a*x])*ArcSin[a*x]^(3/2)) - (I*E^((3*I)*ArcSin[a*x])*(I - 6*ArcSin[a*x]) - 6*Sqrt[3]*((-I)*ArcS
in[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcSin[a*x]])/(12*ArcSin[a*x]^(3/2)) + (1 - (6*I)*ArcSin[a*x] + 6*Sqrt[3]*E^(
(3*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcSin[a*x]])/(12*E^((3*I)*ArcSin[a*x])*ArcSin[a*x]^
(3/2)))/a^3

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Maple [A]  time = 0.055, size = 117, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,{a}^{3}} \left ( -6\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}+2\,\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}-2\,ax\arcsin \left ( ax \right ) +6\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) +\sqrt{-{a}^{2}{x}^{2}+1}-\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/a^3*(-6*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)+2
*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(3/2)-2*a*x*arcsin(a*x)+6*arcsin(a*
x)*sin(3*arcsin(a*x))+(-a^2*x^2+1)^(1/2)-cos(3*arcsin(a*x)))/arcsin(a*x)^(3/2)

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

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/asin(a*x)**(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arcsin(a*x)^(5/2), x)