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

Optimal. Leaf size=95 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a^4}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4}-\frac{3 \sqrt{\sin ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)} \]

[Out]

(-3*Sqrt[ArcSin[a*x]])/(32*a^4) + (x^4*Sqrt[ArcSin[a*x]])/4 - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*
x]]])/(64*a^4) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(16*a^4)

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Rubi [A]  time = 0.189416, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4629, 4723, 3312, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a^4}+\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4}-\frac{3 \sqrt{\sin ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[ArcSin[a*x]])/(32*a^4) + (x^4*Sqrt[ArcSin[a*x]])/4 - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*
x]]])/(64*a^4) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(16*a^4)

Rule 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

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])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 x^3 \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)}-\frac{1}{8} a \int \frac{x^4}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}\\ &=-\frac{3 \sqrt{\sin ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a^4}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^4}\\ &=-\frac{3 \sqrt{\sin ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{32 a^4}+\frac{\operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{8 a^4}\\ &=-\frac{3 \sqrt{\sin ^{-1}(a x)}}{32 a^4}+\frac{1}{4} x^4 \sqrt{\sin ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{64 a^4}+\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{16 a^4}\\ \end{align*}

Mathematica [C]  time = 0.0610767, size = 138, normalized size = 1.45 \[ \frac{\sqrt{\sin ^{-1}(a x)} \left (-4 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}(a x)\right )-4 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},4 i \sin ^{-1}(a x)\right )\right )}{128 a^4 \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[ArcSin[a*x]]*(-4*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] - 4*Sqrt[2]*Sqrt[(-I)*ArcSin
[a*x]]*Gamma[3/2, (2*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-4*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a
*x]]*Gamma[3/2, (4*I)*ArcSin[a*x]]))/(128*a^4*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.049, size = 91, normalized size = 1. \begin{align*}{\frac{1}{128\,{a}^{4}} \left ( -\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +4\,\arcsin \left ( ax \right ) \cos \left ( 4\,\arcsin \left ( ax \right ) \right ) -16\,\arcsin \left ( ax \right ) \cos \left ( 2\,\arcsin \left ( ax \right ) \right ) +8\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128/a^4/arcsin(a*x)^(1/2)*(-2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2)
)+4*arcsin(a*x)*cos(4*arcsin(a*x))-16*arcsin(a*x)*cos(2*arcsin(a*x))+8*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2*a
rcsin(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 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 = 1.43165, size = 207, normalized size = 2.18 \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 )}{512 \, 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 )}{512 \, a^{4}} - \frac{\left (i + 1\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \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 )}{64 \, a^{4}} + \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} - \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} + \frac{\sqrt{\arcsin \left (a x\right )} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{64 \, 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]

(1/512*I + 1/512)*sqrt(2)*sqrt(pi)*erf((I - 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 - (1/512*I - 1/512)*sqrt(2)*sqrt
(pi)*erf(-(I + 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 - (1/64*I + 1/64)*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^4
+ (1/64*I - 1/64)*sqrt(pi)*erf(-(I + 1)*sqrt(arcsin(a*x)))/a^4 + 1/64*sqrt(arcsin(a*x))*e^(4*I*arcsin(a*x))/a
^4 - 1/16*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^4 - 1/16*sqrt(arcsin(a*x))*e^(-2*I*arcsin(a*x))/a^4 + 1/64*s
qrt(arcsin(a*x))*e^(-4*I*arcsin(a*x))/a^4