3.1.84 \(\int \text {ArcCos}(a x)^{3/2} \, dx\) [84]

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

[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

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Rubi [A]
time = 0.08, 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 = {4716, 4768, 4720, 3386, 3432} \begin {gather*} -\frac {3 \sqrt {1-a^2 x^2} \sqrt {\text {ArcCos}(a x)}}{2 a}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{2 a}+x \text {ArcCos}(a x)^{3/2} \end {gather*}

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 3386

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 3432

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

Rule 4716

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 4720

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

Rule 4768

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 \text {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 \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.02, size = 66, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},-i \text {ArcCos}(a x)\right )+\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {5}{2},i \text {ArcCos}(a x)\right )}{2 a \sqrt {\text {ArcCos}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[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]])

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Maple [A]
time = 0.16, size = 72, normalized size = 0.96

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +3 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}-3 \pi \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{4 a \sqrt {\pi }}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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)

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

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)

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Giac [C] Result contains complex when optimal does not.
time = 0.49, size = 119, normalized size = 1.59 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (a x\right )\right )}}{2 \, a} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \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 {\arccos \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 {\arccos \left (a x\right )}\right )}{16 \, a} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (i \, \arccos \left (a x\right )\right )}}{4 \, a} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-i \, \arccos \left (a x\right )\right )}}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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