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

Optimal. Leaf size=121 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{80 a^5}+\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)} \]

[Out]

(x^5*Sqrt[ArcSin[a*x]])/5 - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5) + (Sqrt[Pi/6]*FresnelS
[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(80*a^5)

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Rubi [A]  time = 0.241509, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4629, 4723, 3312, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{80 a^5}+\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^5*Sqrt[ArcSin[a*x]])/5 - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5) + (Sqrt[Pi/6]*FresnelS
[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(80*a^5)

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 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 x^4 \sqrt{\sin ^{-1}(a x)} \, dx &=\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)}-\frac{1}{10} a \int \frac{x^5}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin ^5(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sin (x)}{8 \sqrt{x}}-\frac{5 \sin (3 x)}{16 \sqrt{x}}+\frac{\sin (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{10 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{160 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{80 a^5}+\frac{\operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a^5}-\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{8 a^5}\\ &=\frac{1}{5} x^5 \sqrt{\sin ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{8 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^5}-\frac{\sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{80 a^5}\\ \end{align*}

Mathematica [C]  time = 0.0990047, size = 204, normalized size = 1.69 \[ \frac{i \sqrt{\sin ^{-1}(a x)} \left (-150 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-i \sin ^{-1}(a x)\right )+150 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},i \sin ^{-1}(a x)\right )+25 \sqrt{3} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-3 i \sin ^{-1}(a x)\right )-25 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},3 i \sin ^{-1}(a x)\right )-3 \sqrt{5} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-5 i \sin ^{-1}(a x)\right )+3 \sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},5 i \sin ^{-1}(a x)\right )\right )}{2400 a^5 \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/2400)*Sqrt[ArcSin[a*x]]*(-150*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-I)*ArcSin[a*x]] + 150*Sqrt[(-I)*ArcSin[a*x]
]*Gamma[3/2, I*ArcSin[a*x]] + 25*Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-3*I)*ArcSin[a*x]] - 25*Sqrt[3]*Sqrt[
(-I)*ArcSin[a*x]]*Gamma[3/2, (3*I)*ArcSin[a*x]] - 3*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-5*I)*ArcSin[a*x]]
+ 3*Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (5*I)*ArcSin[a*x]]))/(a^5*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.074, size = 143, normalized size = 1.2 \begin{align*} -{\frac{1}{2400\,{a}^{5}} \left ( 3\,\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -25\,\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 ) +150\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -300\,ax\arcsin \left ( ax \right ) +150\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) -30\,\arcsin \left ( ax \right ) \sin \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)^(1/2),x)

[Out]

-1/2400/a^5/arcsin(a*x)^(1/2)*(3*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))-25*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))+150*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-300*a*x*arcsin(a*x
)+150*arcsin(a*x)*sin(3*arcsin(a*x))-30*arcsin(a*x)*sin(5*arcsin(a*x)))

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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 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 = 1.3452, size = 333, normalized size = 2.75 \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 )}{3200 \, 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 )}{3200 \, 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 )}{384 \, 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 )}{384 \, 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 )}{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 )}{64 \, a^{5}} - \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} + \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} - \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{160 \, 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]

-(1/3200*I - 1/3200)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 + (1/3200*I + 1/3200)
*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 + (1/384*I - 1/384)*sqrt(6)*sqrt(pi)*erf
((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 - (1/384*I + 1/384)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*
sqrt(arcsin(a*x)))/a^5 - (1/64*I - 1/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + (
1/64*I + 1/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 - 1/160*I*sqrt(arcsin(a*x))*
e^(5*I*arcsin(a*x))/a^5 + 1/32*I*sqrt(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^5 - 1/16*I*sqrt(arcsin(a*x))*e^(I*arc
sin(a*x))/a^5 + 1/16*I*sqrt(arcsin(a*x))*e^(-I*arcsin(a*x))/a^5 - 1/32*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x)
)/a^5 + 1/160*I*sqrt(arcsin(a*x))*e^(-5*I*arcsin(a*x))/a^5