3.52 \(\int \frac {\cos ^{-1}(a x^2)}{x^2} \, dx\)

Optimal. Leaf size=29 \[ -\frac {\cos ^{-1}\left (a x^2\right )}{x}-2 \sqrt {a} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right ) \]

[Out]

-arccos(a*x^2)/x-2*EllipticF(x*a^(1/2),I)*a^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4843, 12, 221} \[ -\frac {\cos ^{-1}\left (a x^2\right )}{x}-2 \sqrt {a} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x^2]/x^2,x]

[Out]

-(ArcCos[a*x^2]/x) - 2*Sqrt[a]*EllipticF[ArcSin[Sqrt[a]*x], -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 4843

Int[((a_.) + ArcCos[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcCos[
u]))/(d*(m + 1)), x] + Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

\begin {align*} \int \frac {\cos ^{-1}\left (a x^2\right )}{x^2} \, dx &=-\frac {\cos ^{-1}\left (a x^2\right )}{x}-\int \frac {2 a}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {\cos ^{-1}\left (a x^2\right )}{x}-(2 a) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {\cos ^{-1}\left (a x^2\right )}{x}-2 \sqrt {a} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.04, size = 40, normalized size = 1.38 \[ -\frac {\cos ^{-1}\left (a x^2\right )+2 i \sqrt {-a} x F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x^2]/x^2,x]

[Out]

-((ArcCos[a*x^2] + (2*I)*Sqrt[-a]*x*EllipticF[I*ArcSinh[Sqrt[-a]*x], -1])/x)

________________________________________________________________________________________

fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x^{2}\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(a*x^2)/x^2, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arccos \left (a x^{2}\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccos(a*x^2)/x^2, x)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 57, normalized size = 1.97 \[ -\frac {\arccos \left (a \,x^{2}\right )}{x}-\frac {2 \sqrt {a}\, \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \EllipticF \left (x \sqrt {a}, i\right )}{\sqrt {-a^{2} x^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arccos(a*x^2)/x-2*a^(1/2)*(-a*x^2+1)^(1/2)*(a*x^2+1)^(1/2)/(-a^2*x^4+1)^(1/2)*EllipticF(x*a^(1/2),I)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, a x \int \frac {\sqrt {-a x^{2} + 1}}{\sqrt {a x^{2} + 1} {\left (a x^{2} - 1\right )}}\,{d x} - \arctan \left (\sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}, a x^{2}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*x*integrate(e^(1/2*log(a*x^2 + 1) + 1/2*log(-a*x^2 + 1))/(a^4*x^8 - a^2*x^4 + (a^2*x^4 - 1)*e^(log(a*x^2
+ 1) + log(-a*x^2 + 1))), x) - arctan2(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1), a*x^2))/x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acos}\left (a\,x^2\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x^2)/x^2, x)

________________________________________________________________________________________

sympy [A]  time = 1.07, size = 44, normalized size = 1.52 \[ - \frac {a x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {a^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} - \frac {\operatorname {acos}{\left (a x^{2} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), a**2*x**4*exp_polar(2*I*pi))/(2*gamma(5/4)) - acos(a*x**2)/x

________________________________________________________________________________________