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

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

[Out]

-1/128*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4+1/16*FresnelC(2*arcsin(a*x)^(1/2)/P
i^(1/2))*Pi^(1/2)/a^4-3/32*arcsin(a*x)^(1/2)/a^4+1/4*x^4*arcsin(a*x)^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 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 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 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 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^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*}

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Mathematica [C]  time = 0.06, size = 138, normalized size = 1.45 \[ \frac {\sqrt {\sin ^{-1}(a x)} \left (-4 \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 i \sin ^{-1}(a x)\right )-4 \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 i \sin ^{-1}(a x)\right )+\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},4 i \sin ^{-1}(a x)\right )\right )}{128 a^4 \sqrt {\sin ^{-1}(a x)^2}} \]

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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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^3*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 = 0.53, size = 153, normalized size = 1.61 \[ \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{512 \, a^{4}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{512 \, a^{4}} - \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} \]

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

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maple [A]  time = 0.08, size = 90, normalized size = 0.95 \[ -\frac {\FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+16 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-8 \FresnelC \left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }-4 \arcsin \left (a x \right ) \cos \left (4 \arcsin \left (a x \right )\right )}{128 a^{4} \sqrt {\arcsin \left (a x \right )}} \]

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)*(FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2
)+16*arcsin(a*x)*cos(2*arcsin(a*x))-8*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*arcsin(a*x)^(1/2)*Pi^(1/2)-4*arcs
in(a*x)*cos(4*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^3*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 \[ \int x^3\,\sqrt {\mathrm {asin}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*asin(a*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \]

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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