3.282 \(\int \frac {x^3}{(1-a^2 x^2)^2 \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=43 \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^4}-\frac {\text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)},x\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]

[Out]

Defer[Int][x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]), x]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]),x]

[Out]

Integrate[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]), x]

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-a^2*x^2+1)^2/arctanh(a*x),x, algorithm="fricas")

[Out]

integral(x^3/((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-a^2*x^2+1)^2/arctanh(a*x),x, algorithm="giac")

[Out]

integrate(x^3/((a^2*x^2 - 1)^2*arctanh(a*x)), x)

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maple [A]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-a^2*x^2+1)^2/arctanh(a*x),x)

[Out]

int(x^3/(-a^2*x^2+1)^2/arctanh(a*x),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-a^2*x^2+1)^2/arctanh(a*x),x, algorithm="maxima")

[Out]

integrate(x^3/((a^2*x^2 - 1)^2*arctanh(a*x)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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