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]

-1/800*FresnelS(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+1/96*FresnelS(6^(1/2)/Pi^(1/2)*arcs
in(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5-1/16*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5+1/5
*x^5*arcsin(a*x)^(1/2)

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Rubi [A]  time = 0.24, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 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 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 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]

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])

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*}

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Mathematica [C]  time = 0.10, size = 204, normalized size = 1.69 \[ \frac {i \sqrt {\sin ^{-1}(a x)} \left (-150 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-i \sin ^{-1}(a x)\right )+150 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},i \sin ^{-1}(a x)\right )+25 \sqrt {3} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-3 i \sin ^{-1}(a x)\right )-25 \sqrt {3} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},3 i \sin ^{-1}(a x)\right )-3 \sqrt {5} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-5 i \sin ^{-1}(a x)\right )+3 \sqrt {5} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},5 i \sin ^{-1}(a x)\right )\right )}{2400 a^5 \sqrt {\sin ^{-1}(a x)^2}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [C]  time = 0.30, size = 247, normalized size = 2.04 \[ -\frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{3200 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{3200 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{384 \, a^{5}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{384 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{5}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} \]

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

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maple [A]  time = 0.13, size = 143, normalized size = 1.18 \[ -\frac {3 \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-25 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+150 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-300 a x \arcsin \left (a x \right )+150 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-30 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )}{2400 a^{5} \sqrt {\arcsin \left (a x \right )}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\sqrt {\mathrm {asin}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*asin(a*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \]

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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