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

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{x \cos ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable[1/(x*ArcCos[a*x]^2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0119504, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \cos ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*ArcCos[a*x]^2),x]

[Out]

Defer[Int][1/(x*ArcCos[a*x]^2), x]

Rubi steps

\begin{align*} \int \frac{1}{x \cos ^{-1}(a x)^2} \, dx &=\int \frac{1}{x \cos ^{-1}(a x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.960006, size = 0, normalized size = 0. \[ \int \frac{1}{x \cos ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*ArcCos[a*x]^2),x]

[Out]

Integrate[1/(x*ArcCos[a*x]^2), x]

________________________________________________________________________________________

Maple [A]  time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( \arccos \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/arccos(a*x)^2,x)

[Out]

int(1/x/arccos(a*x)^2,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right ) \int \frac{\sqrt{-a x + 1}}{\sqrt{a x + 1} a x^{3} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right ) - \sqrt{a x + 1} x^{2} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )}\,{d x} - \sqrt{a x + 1} \sqrt{-a x + 1}}{a x \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arccos(a*x)^2,x, algorithm="maxima")

[Out]

-(a*x*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^3*x^4 - a*x^2)*arc
tan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) - sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*x*arctan2(sqrt(a*x + 1)*sqrt(-
a*x + 1), a*x))

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \arccos \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arccos(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/(x*arccos(a*x)^2), x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \operatorname{acos}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/acos(a*x)**2,x)

[Out]

Integral(1/(x*acos(a*x)**2), x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \arccos \left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arccos(a*x)^2,x, algorithm="giac")

[Out]

integrate(1/(x*arccos(a*x)^2), x)