3.101 \(\int \frac{x^4}{\sin ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

[Out]

(-2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^5)
+ (3*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5) - (Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sq
rt[ArcSin[a*x]]])/(4*a^5)

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Rubi [A]  time = 0.098155, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4631, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcSin[a*x]^(3/2),x]

[Out]

(-2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^5)
+ (3*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5) - (Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sq
rt[ArcSin[a*x]]])/(4*a^5)

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]

Rubi steps

\begin{align*} \int \frac{x^4}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{8 \sqrt{x}}+\frac{9 \sin (3 x)}{16 \sqrt{x}}-\frac{5 \sin (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^5}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{5 \operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}+\frac{9 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}\\ \end{align*}

Mathematica [C]  time = 0.151671, size = 319, normalized size = 2.35 \[ \frac{-\frac{e^{i \sin ^{-1}(a x)}-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )}{8 \sqrt{\sin ^{-1}(a x)}}-\frac{e^{-i \sin ^{-1}(a x)}-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )}{8 \sqrt{\sin ^{-1}(a x)}}+\frac{3 \left (e^{3 i \sin ^{-1}(a x)}-\sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )\right )}{16 \sqrt{\sin ^{-1}(a x)}}+\frac{3 \left (e^{-3 i \sin ^{-1}(a x)}-\sqrt{3} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )\right )}{16 \sqrt{\sin ^{-1}(a x)}}-\frac{e^{5 i \sin ^{-1}(a x)}-\sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 i \sin ^{-1}(a x)\right )}{16 \sqrt{\sin ^{-1}(a x)}}-\frac{e^{-5 i \sin ^{-1}(a x)}-\sqrt{5} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 i \sin ^{-1}(a x)\right )}{16 \sqrt{\sin ^{-1}(a x)}}}{a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/ArcSin[a*x]^(3/2),x]

[Out]

(-(E^(I*ArcSin[a*x]) - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]])/(8*Sqrt[ArcSin[a*x]]) - (E^((-I)*A
rcSin[a*x]) - Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin[a*x]])/(8*Sqrt[ArcSin[a*x]]) + (3*(E^((3*I)*ArcSin[a*x])
- Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]]))/(16*Sqrt[ArcSin[a*x]]) + (3*(E^((-3*I)*ArcS
in[a*x]) - Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (3*I)*ArcSin[a*x]]))/(16*Sqrt[ArcSin[a*x]]) - (E^((5*I)*ArcS
in[a*x]) - Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-5*I)*ArcSin[a*x]])/(16*Sqrt[ArcSin[a*x]]) - (E^((-5*I)*
ArcSin[a*x]) - Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (5*I)*ArcSin[a*x]])/(16*Sqrt[ArcSin[a*x]]))/a^5

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Maple [A]  time = 0.051, size = 138, normalized size = 1. \begin{align*} -{\frac{1}{8\,{a}^{5}} \left ( \sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{5}\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) -3\,\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +2\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +2\,\sqrt{-{a}^{2}{x}^{2}+1}-3\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) +\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsin(a*x)^(3/2),x)

[Out]

-1/8/a^5*(5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))-3*3^
(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+2*2^(1/2)*arcsin
(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))+2*(-a^2*x^2+1)^(1/2)-3*cos(3*arcsin(a*x))+co
s(5*arcsin(a*x)))/arcsin(a*x)^(1/2)

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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^4/arcsin(a*x)^(3/2),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^4/arcsin(a*x)^(3/2),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^{4}}{\operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asin(a*x)**(3/2),x)

[Out]

Integral(x**4/asin(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\arcsin \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^4/arcsin(a*x)^(3/2), x)