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

Optimal. Leaf size=65 \[ \frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4}-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^4} \]

[Out]

-(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^4) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt
[Pi]])/(4*a^4)

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Rubi [A]  time = 0.0826256, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4635, 4406, 3305, 3351} \[ \frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4}-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^4) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt
[Pi]])/(4*a^4)

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

Mathematica [C]  time = 0.0323625, size = 128, normalized size = 1.97 \[ \frac{-2 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )-2 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \sin ^{-1}(a x)\right )}{32 a^4 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (
2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (4*
I)*ArcSin[a*x]])/(32*a^4*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.039, size = 44, normalized size = 0.7 \begin{align*}{\frac{\sqrt{\pi }}{16\,{a}^{4}} \left ( -\sqrt{2}{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +4\,{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arcsin(a*x)^(1/2),x)

[Out]

1/16/a^4*Pi^(1/2)*(-2^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))+4*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(
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^3/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^3/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^{3}}{\sqrt{\operatorname{asin}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/asin(a*x)**(1/2),x)

[Out]

Integral(x**3/sqrt(asin(a*x)), x)

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Giac [C]  time = 1.47587, size = 109, normalized size = 1.68 \begin{align*} -\frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{64 \, a^{4}} + \frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{64 \, a^{4}} + \frac{\left (i - 1\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{16 \, a^{4}} - \frac{\left (i + 1\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{16 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arcsin(a*x)^(1/2),x, algorithm="giac")

[Out]

-(1/64*I - 1/64)*sqrt(2)*sqrt(pi)*erf((I - 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 + (1/64*I + 1/64)*sqrt(2)*sqrt(pi
)*erf(-(I + 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 + (1/16*I - 1/16)*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^4 -
(1/16*I + 1/16)*sqrt(pi)*erf(-(I + 1)*sqrt(arcsin(a*x)))/a^4