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

Optimal. Leaf size=175 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2} \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+\text{Shi}\left (4 \tanh ^{-1}(a x)\right )-\frac{1}{2 a x \tanh ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.766691, 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 = {} \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-1/(2*a*x*ArcTanh[a*x]^2) - (a*x)/(2*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) - (a*x)/(2*(1 - a^2*x^2)*ArcTanh[a*x]^2)
- 2/((1 - a^2*x^2)^2*ArcTanh[a*x]) + 3/(2*(1 - a^2*x^2)*ArcTanh[a*x]) - (1 + a^2*x^2)/(2*(1 - a^2*x^2)*ArcTanh
[a*x]) + (3*SinhIntegral[2*ArcTanh[a*x]])/2 + SinhIntegral[4*ArcTanh[a*x]] - 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 )^3 \tanh ^{-1}(a x)^3} \, dx &=a^2 \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx+\int \frac{1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2} a \int \frac{1}{\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 )^2 \tanh ^{-1}(a x)^3} \, dx+\frac{1}{2} \left (3 a^3\right ) \int \frac{x^2}{\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 ) \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\frac{1}{2} (3 a) \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx-\frac{1}{2} (3 a) \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\left (2 a^2\right ) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\left (2 a^2\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+2 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}-\left (3 a^2\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\left (6 a^2\right ) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )+2 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+6 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )+6 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{3}{2} \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{4} \text{Shi}\left (4 \tanh ^{-1}(a x)\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac{1}{2 a x \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{3}{2} \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+\text{Shi}\left (4 \tanh ^{-1}(a x)\right )-\frac{\int \frac{1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ \end{align*}

Mathematica [A]  time = 5.38553, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.221, 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 ) ^{3}}}\, 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)^3,x)

[Out]

int(1/x/(-a^2*x^2+1)^3/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 (5 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) -{\left (5 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{{\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \,{\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) +{\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (-a x + 1\right )^{2}} + \int -\frac{2 \,{\left (10 \, a^{4} x^{4} - 3 \, a^{2} x^{2} + 1\right )}}{{\left (a^{8} x^{9} - 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} - a^{2} x^{3}\right )} \log \left (a x + 1\right ) -{\left (a^{8} x^{9} - 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} - a^{2} x^{3}\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)^3,x, algorithm="maxima")

[Out]

-(2*a*x + (5*a^2*x^2 - 1)*log(a*x + 1) - (5*a^2*x^2 - 1)*log(-a*x + 1))/((a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*log(a
*x + 1)^2 - 2*(a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*log(a*x + 1)*log(-a*x + 1) + (a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*log
(-a*x + 1)^2) + integrate(-2*(10*a^4*x^4 - 3*a^2*x^2 + 1)/((a^8*x^9 - 3*a^6*x^7 + 3*a^4*x^5 - a^2*x^3)*log(a*x
+ 1) - (a^8*x^9 - 3*a^6*x^7 + 3*a^4*x^5 - 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^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, 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)^3/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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