3.92 \(\int \frac{x^4}{\sqrt{\sin ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=106 \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5}+\frac{\sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5} \]

[Out]

(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5) - (Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[
a*x]]])/(8*a^5) + (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5)

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Rubi [A]  time = 0.111608, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4635, 4406, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4 a^5}-\frac{\sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5}+\frac{\sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[ArcSin[a*x]],x]

[Out]

(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5) - (Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[
a*x]]])/(8*a^5) + (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5)

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]

Rubi steps

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

Mathematica [C]  time = 0.0565194, size = 193, normalized size = 1.82 \[ -\frac{i \left (10 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )-10 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )-5 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )+5 \sqrt{3} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )+\sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 i \sin ^{-1}(a x)\right )-\sqrt{5} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 i \sin ^{-1}(a x)\right )\right )}{160 a^5 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/Sqrt[ArcSin[a*x]],x]

[Out]

((-I/160)*(10*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]] - 10*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin
[a*x]] - 5*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-3*I)*ArcSin[a*x]] + 5*Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma
[1/2, (3*I)*ArcSin[a*x]] + Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-5*I)*ArcSin[a*x]] - Sqrt[5]*Sqrt[I*ArcS
in[a*x]]*Gamma[1/2, (5*I)*ArcSin[a*x]]))/(a^5*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.055, size = 72, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{80\,{a}^{5}} \left ( \sqrt{5}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{5}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) -5\,\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +10\,{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arcsin(a*x)^(1/2),x)

[Out]

1/80/a^5*2^(1/2)*Pi^(1/2)*(5^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))-5*3^(1/2)*FresnelC(2^(
1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+10*FresnelC(2^(1/2)/Pi^(1/2)*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)^(1/2),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)^(1/2),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}}{\sqrt{\operatorname{asin}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/asin(a*x)**(1/2),x)

[Out]

Integral(x**4/sqrt(asin(a*x)), x)

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Giac [C]  time = 1.45233, size = 188, normalized size = 1.77 \begin{align*} -\frac{\left (i + 1\right ) \, \sqrt{10} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{10} \sqrt{\arcsin \left (a x\right )}\right )}{320 \, a^{5}} + \frac{\left (i - 1\right ) \, \sqrt{10} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{10} \sqrt{\arcsin \left (a x\right )}\right )}{320 \, a^{5}} + \frac{\left (i + 1\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{6} \sqrt{\arcsin \left (a x\right )}\right )}{64 \, a^{5}} - \frac{\left (i - 1\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{6} \sqrt{\arcsin \left (a x\right )}\right )}{64 \, a^{5}} - \frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{32 \, a^{5}} + \frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{32 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arcsin(a*x)^(1/2),x, algorithm="giac")

[Out]

-(1/320*I + 1/320)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 + (1/320*I - 1/320)*sqr
t(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 + (1/64*I + 1/64)*sqrt(6)*sqrt(pi)*erf((1/2*
I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 - (1/64*I - 1/64)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arc
sin(a*x)))/a^5 - (1/32*I + 1/32)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + (1/32*I -
1/32)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5