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

Optimal. Leaf size=98 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2} \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{2} \text{Chi}\left (4 \tanh ^{-1}(a x)\right )-\frac{1}{a x \tanh ^{-1}(a x)} \]

[Out]

-(1/(a*x*ArcTanh[a*x])) - (a*x)/((1 - a^2*x^2)^2*ArcTanh[a*x]) - (a*x)/((1 - a^2*x^2)*ArcTanh[a*x]) + (3*CoshI
ntegral[2*ArcTanh[a*x]])/2 + CoshIntegral[4*ArcTanh[a*x]]/2 - Unintegrable[1/(x^2*ArcTanh[a*x]), x]/a

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Rubi [A]  time = 0.650164, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a*x*ArcTanh[a*x])) - (a*x)/((1 - a^2*x^2)^2*ArcTanh[a*x]) - (a*x)/((1 - a^2*x^2)*ArcTanh[a*x]) + (3*CoshI
ntegral[2*ArcTanh[a*x]])/2 + CoshIntegral[4*ArcTanh[a*x]]/2 - Defer[Int][1/(x^2*ArcTanh[a*x]), x]/a

Rubi steps

\begin{align*} \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx &=a^2 \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+a \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+a^2 \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\left (3 a^3\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\int \frac{1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{a x \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+3 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}+a \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+a^3 \int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{1}{a x \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+3 \operatorname{Subst}\left (\int \left (-\frac{1}{8 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}+\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{1}{a x \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac{3}{8} \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}-\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{1}{a x \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1}{2} \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{2} \text{Chi}\left (4 \tanh ^{-1}(a x)\right )+2 \left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ &=-\frac{1}{a x \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{3}{2} \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{2} \text{Chi}\left (4 \tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ \end{align*}

Mathematica [A]  time = 4.26807, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.249, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, 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)^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2}{{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (-a x + 1\right )} + \int -\frac{2 \,{\left (5 \, a^{2} x^{2} - 1\right )}}{{\left (a^{7} x^{8} - 3 \, a^{5} x^{6} + 3 \, a^{3} x^{4} - a x^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{7} x^{8} - 3 \, a^{5} x^{6} + 3 \, a^{3} x^{4} - a x^{2}\right )} \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)^3/arctanh(a*x)^2,x, algorithm="maxima")

[Out]

-2/((a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) - (a^5*x^5 - 2*a^3*x^3 + a*x)*log(-a*x + 1)) + integrate(-2*(5*a^
2*x^2 - 1)/((a^7*x^8 - 3*a^5*x^6 + 3*a^3*x^4 - a*x^2)*log(a*x + 1) - (a^7*x^8 - 3*a^5*x^6 + 3*a^3*x^4 - a*x^2)
*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^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )} \operatorname{artanh}\left (a x\right )^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(-1/((a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x)*arctanh(a*x)^2), 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}^{2}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname{atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname{atanh}^{2}{\left (a x \right )} - x \operatorname{atanh}^{2}{\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)**2,x)

[Out]

-Integral(1/(a**6*x**7*atanh(a*x)**2 - 3*a**4*x**5*atanh(a*x)**2 + 3*a**2*x**3*atanh(a*x)**2 - x*atanh(a*x)**2
), 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 )^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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