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

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

[Out]

1/80*FresnelC(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+1/8*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(
a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5-1/16*FresnelC(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5

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Rubi [A]  time = 0.11, antiderivative size = 106, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.05, size = 193, normalized size = 1.82 \[ -\frac {i \left (10 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )-10 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )-5 \sqrt {3} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \sin ^{-1}(a x)\right )+5 \sqrt {3} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \sin ^{-1}(a x)\right )+\sqrt {5} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-5 i \sin ^{-1}(a x)\right )-\sqrt {5} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},5 i \sin ^{-1}(a x)\right )\right )}{160 a^5 \sqrt {\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1/160*I)*(10*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]] - 10*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcS
in[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]]*Gam
ma[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*Ar
cSin[a*x]]*Gamma[1/2, (5*I)*ArcSin[a*x]]))/(a^5*Sqrt[ArcSin[a*x]])

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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 = 0.56, size = 139, normalized size = 1.31 \[ -\frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{320 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{320 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{5}} - \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{32 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{32 \, a^{5}} \]

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

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maple [A]  time = 0.10, size = 72, normalized size = 0.68 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (-5 \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+\sqrt {5}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+10 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{80 a^{5}} \]

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*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+5^(1/2)*FresnelC(2^
(1/2)/Pi^(1/2)*5^(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 = 0.00, 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 \[ \int \frac {x^4}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \]

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 \[ \int \frac {x^{4}}{\sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \]

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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