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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (1-a^2 x^2\right )^2}{x \tanh ^{-1}(a x)^2},x\right ) \]

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-a^2 x^2\right )^2}{x \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}{x \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.91, size = 0, normalized size = 0.00 \[ \int \frac {\left (-a^{2} x^{2}+1\right )^{2}}{x \arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )}}{a x \log \left (a x + 1\right ) - a x \log \left (-a x + 1\right )} + \int -\frac {2 \, {\left (5 \, a^{6} x^{6} - 9 \, a^{4} x^{4} + 3 \, a^{2} x^{2} + 1\right )}}{a x^{2} \log \left (a x + 1\right ) - a x^{2} \log \left (-a x + 1\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)/(a*x*log(a*x + 1) - a*x*log(-a*x + 1)) + integrate(-2*(5*a^6*x^6 - 9*a
^4*x^4 + 3*a^2*x^2 + 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 = -0.04 \[ \int \frac {{\left (a^2\,x^2-1\right )}^2}{x\,{\mathrm {atanh}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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