3.3.52 \(\int \frac {x}{(1-a^2 x^2) \tanh ^{-1}(a x)^2} \, dx\) [252]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {} \begin {gather*} \int \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \end {gather*}

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 = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \end {gather*}

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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Maple [A]
time = 4.13, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \end {gather*}

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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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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