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

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

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 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 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 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 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]

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*}

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Mathematica [C]  time = 0.04, size = 76, normalized size = 1.01 \[ \frac {\sqrt {\sin ^{-1}(a x)} \left (\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},i \sin ^{-1}(a x)\right )\right )}{2 a \sqrt {\sin ^{-1}(a x)^2}} \]

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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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(arcsin(a*x)^(3/2),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 = 0.33, size = 119, normalized size = 1.59 \[ -\frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a} - \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{4 \, a} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{4 \, a} \]

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

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maple [A]  time = 0.05, size = 72, normalized size = 0.96 \[ \frac {\sqrt {2}\, \left (2 \arcsin \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x a +3 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}-3 \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{4 a \sqrt {\pi }} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^(3/2),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 \[ \int {\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(asin(a*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

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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