3.3.82 \(\int \frac {x^3}{(1-a^2 x^2)^2 \tanh ^{-1}(a x)} \, dx\) [282]

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 = 0.04, 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 = {} \begin {gather*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \end {gather*}

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 = 1.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx \end {gather*}

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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Maple [A]
time = 37.94, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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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