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

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

[Out]

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

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Rubi [A]  time = 0.014442, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 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]

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]

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 = 0.0385381, size = 40, normalized size = 1.38 \[ -\frac{\cos ^{-1}\left (a x^2\right )+2 i \sqrt{-a} x \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )}{x} \]

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)

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Maple [B]  time = 0.006, size = 57, normalized size = 2. \begin{align*} -{\frac{\arccos \left ( a{x}^{2} \right ) }{x}}-2\,{\frac{\sqrt{a}\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{a},i \right ) }{\sqrt{-{a}^{2}{x}^{4}+1}}} \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arccos \left (a x^{2}\right )}{x^{2}}, x\right ) \end{align*}

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)

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Sympy [A]  time = 1.21139, size = 44, normalized size = 1.52 \begin{align*} - \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} \end{align*}

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arccos \left (a x^{2}\right )}{x^{2}}\,{d x} \end{align*}

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)