3.3.92 \(\int \frac {x^3}{(1-a^2 x^2)^2 \tanh ^{-1}(a x)^3} \, dx\) [292]

Optimal. Leaf size=102 \[ \frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)^2},x\right )}{2 a^3} \]

[Out]

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

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Rubi [A]
time = 0.17, 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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x/(2*a^3*ArcTanh[a*x]^2) - x/(2*a^3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^4*(1 - a^2*x^2)*ArcTanh
[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^4 - Defer[Int][ArcTanh[a*x]^(-2), x]/(2*a^3)

Rubi steps

\begin {align*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx}{a^2}-\frac {\int \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=\frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\int \frac {1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}+\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^2}\\ &=\frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac {\int \frac {1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac {\int \frac {1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac {\int \frac {1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac {\int \frac {1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ \end {align*}

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Mathematica [A]
time = 7.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 29.26, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

(2*a*x^3 - (a^2*x^4 - 3*x^2)*log(a*x + 1) + (a^2*x^4 - 3*x^2)*log(-a*x + 1))/((a^4*x^2 - a^2)*log(a*x + 1)^2 -
2*(a^4*x^2 - a^2)*log(a*x + 1)*log(-a*x + 1) + (a^4*x^2 - a^2)*log(-a*x + 1)^2) - integrate(-2*(a^4*x^5 - 2*a
^2*x^3 + 3*x)/((a^6*x^4 - 2*a^4*x^2 + a^2)*log(a*x + 1) - (a^6*x^4 - 2*a^4*x^2 + a^2)*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^3/(-a^2*x^2+1)^2/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\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)^3*(a^2*x^2 - 1)^2),x)

[Out]

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

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