3.115 \(\int \frac{x^2}{\sin ^{-1}(a x)^{7/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}-\frac{6 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}} \]

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (8*x)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^3)/(5*ArcSin[a*x]
^(3/2)) - (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcSin[a*x]]) + (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcSin[a*x]
]) + (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^3) - (6*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[A
rcSin[a*x]]])/(5*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.347694, antiderivative size = 191, 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, 4631, 3305, 3351, 4621, 4723} \[ \frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}-\frac{6 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSin[a*x]^(7/2),x]

[Out]

(-2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (8*x)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^3)/(5*ArcSin[a*x]
^(3/2)) - (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcSin[a*x]]) + (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcSin[a*x]
]) + (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^3) - (6*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[A
rcSin[a*x]]])/(5*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 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3351

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

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)^{7/2}} \, dx &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac{4 \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-\frac{1}{5} (6 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{12}{5} \int \frac{x^2}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac{8 \int \frac{1}{\sin ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{24 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 \sqrt{x}}+\frac{3 \sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}-\frac{16 \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{15 a}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}-\frac{18 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}-\frac{32 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}+\frac{12 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}-\frac{36 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac{8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac{4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac{16 \sqrt{1-a^2 x^2}}{15 a^3 \sqrt{\sin ^{-1}(a x)}}+\frac{24 x^2 \sqrt{1-a^2 x^2}}{5 a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{15 a^3}-\frac{6 \sqrt{6 \pi } S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{5 a^3}\\ \end{align*}

Mathematica [C]  time = 0.47627, size = 280, normalized size = 1.47 \[ \frac{-4 \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )+36 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (4 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )+4 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)-3\right )-3 e^{-3 i \sin ^{-1}(a x)} \left (12 \sqrt{3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )+12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)-1\right )+3 e^{3 i \sin ^{-1}(a x)} \left (-12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+1\right )+e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )}{60 a^3 \sin ^{-1}(a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/ArcSin[a*x]^(7/2),x]

[Out]

(3*E^((3*I)*ArcSin[a*x])*(1 + (2*I)*ArcSin[a*x] - 12*ArcSin[a*x]^2) + E^(I*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a*x
] + 4*ArcSin[a*x]^2) - 4*Sqrt[(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-I)*ArcSin[a*x]] + (-3 + (2*I)*ArcSi
n[a*x] + 4*ArcSin[a*x]^2 + 4*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2, I*ArcSin[a*x]])/E^(I*ArcSin[a*
x]) + 36*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-3*I)*ArcSin[a*x]] - (3*(-1 + (2*I)*ArcSin[a
*x] + 12*ArcSin[a*x]^2 + 12*Sqrt[3]*E^((3*I)*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)*Gamma[1/2, (3*I)*ArcSin[a*x]])
)/E^((3*I)*ArcSin[a*x]))/(60*a^3*ArcSin[a*x]^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.055, size = 154, normalized size = 0.8 \begin{align*} -{\frac{1}{30\,{a}^{3}} \left ( 36\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}-4\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}-4\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+36\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) -2\,ax\arcsin \left ( ax \right ) +6\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) +3\,\sqrt{-{a}^{2}{x}^{2}+1}-3\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arcsin(a*x)^(7/2),x)

[Out]

-1/30/a^3*(36*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-
4*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*arcsin(a*x)^(5/2)-4*arcsin(a*x)^2*(-a^2*x^2+1)
^(1/2)+36*arcsin(a*x)^2*cos(3*arcsin(a*x))-2*a*x*arcsin(a*x)+6*arcsin(a*x)*sin(3*arcsin(a*x))+3*(-a^2*x^2+1)^(
1/2)-3*cos(3*arcsin(a*x)))/arcsin(a*x)^(5/2)

________________________________________________________________________________________

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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [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**2/asin(a*x)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\arcsin \left (a x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^2/arcsin(a*x)^(7/2), x)