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

Optimal. Leaf size=48 \[ -\text{Unintegrable}\left (\frac{1}{x \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)},x\right )+\frac{3}{4} \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{8} \text{Shi}\left (4 \tanh ^{-1}(a x)\right ) \]

[Out]

(3*SinhIntegral[2*ArcTanh[a*x]])/4 + SinhIntegral[4*ArcTanh[a*x]]/8 - Unintegrable[1/(x*(-1 + a^2*x^2)*ArcTanh
[a*x]), x]

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Rubi [A]  time = 0.0636374, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.23197, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3}{\it Artanh} \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )} \operatorname{artanh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/((a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x)*arctanh(a*x)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a^{6} x^{7} \operatorname{atanh}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname{atanh}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname{atanh}{\left (a x \right )} - x \operatorname{atanh}{\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)**3/atanh(a*x),x)

[Out]

-Integral(1/(a**6*x**7*atanh(a*x) - 3*a**4*x**5*atanh(a*x) + 3*a**2*x**3*atanh(a*x) - x*atanh(a*x)), 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 )}^{3} x \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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