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

Optimal. Leaf size=26 \[ \frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)},x\right )}{a}-\frac {x}{a \tanh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x}{{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (-a^{2} x^{2}+1\right ) \arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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 \[ -\frac {2 \, x}{a \log \left (a x + 1\right ) - a \log \left (-a x + 1\right )} - 2 \, \int -\frac {1}{a \log \left (a x + 1\right ) - a \log \left (-a x + 1\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-int(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 \[ - \int \frac {x}{a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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