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

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

[Out]

x*arccos(a*x)^(3/2)+3/4*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a-3/2*(-a^2*x^2+1)^(1/2)
*arccos(a*x)^(1/2)/a

________________________________________________________________________________________

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 = {4620, 4678, 4624, 3305, 3351} \[ -\frac {3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{2 a}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a}+x \cos ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(2*a) + x*ArcCos[a*x]^(3/2) + (3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[
ArcCos[a*x]]])/(2*a)

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4620

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

Rule 4624

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

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[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*ArcCos[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 \cos ^{-1}(a x)^{3/2} \, dx &=x \cos ^{-1}(a x)^{3/2}+\frac {1}{2} (3 a) \int \frac {x \sqrt {\cos ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{2 a}+x \cos ^{-1}(a x)^{3/2}-\frac {3}{4} \int \frac {1}{\sqrt {\cos ^{-1}(a x)}} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{2 a}+x \cos ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a}\\ &=-\frac {3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{2 a}+x \cos ^{-1}(a x)^{3/2}+\frac {3 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2 a}\\ &=-\frac {3 \sqrt {1-a^2 x^2} \sqrt {\cos ^{-1}(a x)}}{2 a}+x \cos ^{-1}(a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{2 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 66, normalized size = 0.88 \[ -\frac {\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {5}{2},-i \cos ^{-1}(a x)\right )+\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {5}{2},i \cos ^{-1}(a x)\right )}{2 a \sqrt {\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(Sqrt[(-I)*ArcCos[a*x]]*Gamma[5/2, (-I)*ArcCos[a*x]] + Sqrt[I*ArcCos[a*x]]*Gamma[5/2, I*ArcCos[a*x]])/(a*
Sqrt[ArcCos[a*x]])

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

giac [B]  time = 0.33, size = 144, normalized size = 1.92 \[ \frac {3 \, i \sqrt {\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{4 \, a} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \arccos \left (a x\right )\right )}}{2 \, a} - \frac {3 \, i \sqrt {\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{4 \, a} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \arccos \left (a x\right )\right )}}{2 \, a} - \frac {3 \, \sqrt {2} \sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {2} i \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a {\left (i - 1\right )}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a {\left (i - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*i*sqrt(arccos(a*x))*e^(i*arccos(a*x))/a + 1/2*arccos(a*x)^(3/2)*e^(i*arccos(a*x))/a - 3/4*i*sqrt(arccos(a*
x))*e^(-i*arccos(a*x))/a + 1/2*arccos(a*x)^(3/2)*e^(-i*arccos(a*x))/a - 3/8*sqrt(2)*sqrt(pi)*i*erf(-sqrt(2)*i*
sqrt(arccos(a*x))/(i - 1))/(a*(i - 1)) + 3/8*sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a*(i - 1
))

________________________________________________________________________________________

maple [A]  time = 0.13, size = 72, normalized size = 0.96 \[ \frac {\sqrt {2}\, \left (2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x a -3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+3 \pi \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{4 a \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*2^(1/2)/Pi^(1/2)*(2*arccos(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*x*a-3*2^(1/2)*arccos(a*x)^(1/2)*Pi^(1/2)*(-a^2*x^
2+1)^(1/2)+3*Pi*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)))

________________________________________________________________________________________

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(arccos(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.

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**(3/2), x)

________________________________________________________________________________________