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

Optimal. Leaf size=33 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac {1}{a x \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/((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 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )} x \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right ) \arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{2} x^{3} \operatorname {atanh}^{2}{\left (a x \right )} - x \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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