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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {x \left (1-a^2 x^2\right )^2}{\tanh ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {x \left (1-a^2 x^2\right )^2}{\tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {x \left (1-a^2 x^2\right )^2}{\tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} x^{5} - 2 \, a^{2} x^{3} + x}{\operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-a^2*x^2+1)^2/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 = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {x \left (-a^{2} x^{2}+1\right )^{2}}{\arctanh \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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 \[ \frac {2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}{a \log \left (a x + 1\right ) - a \log \left (-a x + 1\right )} + \int -\frac {2 \, {\left (7 \, a^{6} x^{6} - 15 \, a^{4} x^{4} + 9 \, a^{2} x^{2} - 1\right )}}{a \log \left (a x + 1\right ) - a \log \left (-a x + 1\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x)/(a*log(a*x + 1) - a*log(-a*x + 1)) + integrate(-2*(7*a^6*x^6 - 15*a^4*
x^4 + 9*a^2*x^2 - 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 \[ \int \frac {x\,{\left (a^2\,x^2-1\right )}^2}{{\mathrm {atanh}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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