3.485 \(\int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2, x]

________________________________________________________________________________________

Rubi [A]  time = 0.0302258, 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 = {} \[ \int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2,x]

[Out]

Defer[Int][Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2, x]

Rubi steps

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

Mathematica [A]  time = 1.46264, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-a^2 x^2}}{\tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2,x]

[Out]

Integrate[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2, x]

________________________________________________________________________________________

Maple [A]  time = 0.247, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^(1/2)/arctanh(a*x)^2,x)

[Out]

int((-a^2*x^2+1)^(1/2)/arctanh(a*x)^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^(1/2)/arctanh(a*x)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/arctanh(a*x)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^(1/2)/arctanh(a*x)^2,x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)/arctanh(a*x)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**(1/2)/atanh(a*x)**2,x)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))/atanh(a*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^(1/2)/arctanh(a*x)^2,x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)/arctanh(a*x)^2, x)