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

Optimal. Leaf size=59 \[ \frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^2}-\frac {\sqrt {\sin ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sin ^{-1}(a x)} \]

[Out]

1/8*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^2-1/4*arcsin(a*x)^(1/2)/a^2+1/2*x^2*arcsin(a*x)^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4629, 4723, 3312, 3304, 3352} \[ \frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^2}-\frac {\sqrt {\sin ^{-1}(a x)}}{4 a^2}+\frac {1}{2} x^2 \sqrt {\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[ArcSin[a*x]]/(4*a^2) + (x^2*Sqrt[ArcSin[a*x]])/2 + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(
8*a^2)

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

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Mathematica [C]  time = 0.02, size = 81, normalized size = 1.37 \[ -\frac {\sqrt {\sin ^{-1}(a x)} \left (\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 i \sin ^{-1}(a x)\right )\right )}{8 \sqrt {2} a^2 \sqrt {\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/8*(Sqrt[ArcSin[a*x]]*(Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2
, (2*I)*ArcSin[a*x]]))/(Sqrt[2]*a^2*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*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 = 0.57, size = 71, normalized size = 1.20 \[ -\frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{32 \, a^{2}} + \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{32 \, a^{2}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsin(a*x)^(1/2),x, algorithm="giac")

[Out]

-(1/32*I + 1/32)*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^2 + (1/32*I - 1/32)*sqrt(pi)*erf(-(I + 1)*sqrt(arcs
in(a*x)))/a^2 - 1/8*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^2 - 1/8*sqrt(arcsin(a*x))*e^(-2*I*arcsin(a*x))/a^2

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maple [A]  time = 0.06, size = 42, normalized size = 0.71 \[ \frac {-2 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arcsin \left (a x \right )\right )+\pi \FresnelC \left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )}{8 a^{2} \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/a^2/Pi^(1/2)*(-2*arcsin(a*x)^(1/2)*Pi^(1/2)*cos(2*arcsin(a*x))+Pi*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(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*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 = -0.02 \[ \int x\,\sqrt {\mathrm {asin}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*asin(a*x)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asin(a*x)**(1/2),x)

[Out]

Integral(x*sqrt(asin(a*x)), x)

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