3.84 \(\int \sin ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=75 \[ \frac{3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{2 a}-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a}+x \sin ^{-1}(a x)^{3/2} \]

[Out]

(3*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(2*a) + x*ArcSin[a*x]^(3/2) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[A
rcSin[a*x]]])/(2*a)

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Rubi [A]  time = 0.0991487, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4619, 4677, 4623, 3304, 3352} \[ \frac{3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{2 a}-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a}+x \sin ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^(3/2),x]

[Out]

(3*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(2*a) + x*ArcSin[a*x]^(3/2) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[A
rcSin[a*x]]])/(2*a)

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4623

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[a/b - x/b], x], x, a
+ b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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]

Rubi steps

\begin{align*} \int \sin ^{-1}(a x)^{3/2} \, dx &=x \sin ^{-1}(a x)^{3/2}-\frac{1}{2} (3 a) \int \frac{x \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac{3}{4} \int \frac{1}{\sqrt{\sin ^{-1}(a x)}} \, dx\\ &=\frac{3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a}\\ &=\frac{3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{2 a}\\ &=\frac{3 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a}\\ \end{align*}

Mathematica [C]  time = 0.0433252, size = 76, normalized size = 1.01 \[ \frac{\sqrt{\sin ^{-1}(a x)} \left (\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-i \sin ^{-1}(a x)\right )+\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},i \sin ^{-1}(a x)\right )\right )}{2 a \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[ArcSin[a*x]]*(Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, I*Arc
Sin[a*x]]))/(2*a*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.036, size = 72, normalized size = 1. \begin{align*}{\frac{\sqrt{2}}{4\,a\sqrt{\pi }} \left ( 2\, \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }xa+3\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1}-3\,\pi \,{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^(3/2),x)

[Out]

1/4/a*2^(1/2)*(2*arcsin(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*x*a+3*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2
)-3*Pi*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2)))/Pi^(1/2)

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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(arcsin(a*x)^(3/2),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(arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**(3/2),x)

[Out]

Integral(asin(a*x)**(3/2), x)

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Giac [C]  time = 1.46949, size = 161, normalized size = 2.15 \begin{align*} -\frac{i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac{i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac{\left (3 i + 3\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 )}{16 \, a} - \frac{\left (3 i - 3\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 )}{16 \, a} + \frac{3 \, \sqrt{\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{4 \, a} + \frac{3 \, \sqrt{\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

-1/2*I*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a + 1/2*I*arcsin(a*x)^(3/2)*e^(-I*arcsin(a*x))/a + (3/16*I + 3/16)*
sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a - (3/16*I - 3/16)*sqrt(2)*sqrt(pi)*erf(-(1/2*I
+ 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a + 3/4*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a + 3/4*sqrt(arcsin(a*x))*e^(-I
*arcsin(a*x))/a