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

Optimal. Leaf size=62 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text {Chi}\left (2 \tanh ^{-1}(a x)\right )-\frac {1}{a x \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a*x*ArcTanh[a*x])) - (a*x)/((1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]] - 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 )^2 \tanh ^{-1}(a x)^2} \, dx &=a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, 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 ) \tanh ^{-1}(a x)}-\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\\ &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\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 \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 ) \tanh ^{-1}(a x)}-\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 ) \tanh ^{-1}(a x)}+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 ) \tanh ^{-1}(a x)}+\text {Chi}\left (2 \tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ \end {align*}

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Mathematica [A]  time = 4.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + x\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{2} x \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{{\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right ) - {\left (a^{3} x^{3} - a x\right )} \log \left (-a x + 1\right )} - \int -\frac {2 \, {\left (3 \, a^{2} x^{2} - 1\right )}}{{\left (a^{5} x^{6} - 2 \, a^{3} x^{4} + a x^{2}\right )} \log \left (a x + 1\right ) - {\left (a^{5} x^{6} - 2 \, a^{3} x^{4} + a x^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((a^3*x^3 - a*x)*log(a*x + 1) - (a^3*x^3 - a*x)*log(-a*x + 1)) - integrate(-2*(3*a^2*x^2 - 1)/((a^5*x^6 - 2*
a^3*x^4 + a*x^2)*log(a*x + 1) - (a^5*x^6 - 2*a^3*x^4 + a*x^2)*log(-a*x + 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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