3.257 \(\int \frac{1}{x (1-a^2 x^2) \tanh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a}-\frac{1}{2 a x \tanh ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0784743, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^3),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.721442, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^3),x]

[Out]

Integrate[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^3), x]

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Maple [A]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-a^2*x^2+1)/arctanh(a*x)^3,x)

[Out]

int(1/x/(-a^2*x^2+1)/arctanh(a*x)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a^2*x^2+1)/arctanh(a*x)^3,x, algorithm="maxima")

[Out]

-(2*a*x + (a^2*x^2 - 1)*log(a*x + 1) - (a^2*x^2 - 1)*log(-a*x + 1))/(a^2*x^2*log(a*x + 1)^2 - 2*a^2*x^2*log(a*
x + 1)*log(-a*x + 1) + a^2*x^2*log(-a*x + 1)^2) - 2*integrate(-1/(a^2*x^3*log(a*x + 1) - a^2*x^3*log(-a*x + 1)
), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (a^{2} x^{3} - x\right )} \operatorname{artanh}\left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a^2*x^2+1)/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

integral(-1/((a^2*x^3 - x)*arctanh(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a^{2} x^{3} \operatorname{atanh}^{3}{\left (a x \right )} - x \operatorname{atanh}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a**2*x**2+1)/atanh(a*x)**3,x)

[Out]

-Integral(1/(a**2*x**3*atanh(a*x)**3 - x*atanh(a*x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-a^2*x^2+1)/arctanh(a*x)^3,x, algorithm="giac")

[Out]

integrate(-1/((a^2*x^2 - 1)*x*arctanh(a*x)^3), x)