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

Optimal. Leaf size=99 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {a^2 x^2+1}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text {Shi}\left (2 \tanh ^{-1}(a x)\right )-\frac {1}{2 a x \tanh ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.25, 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 = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.83, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + x\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/((a^4*x^5 - 2*a^2*x^3 + x)*arctanh(a*x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*x + (3*a^2*x^2 - 1)*log(a*x + 1) - (3*a^2*x^2 - 1)*log(-a*x + 1))/((a^4*x^4 - a^2*x^2)*log(a*x + 1)^2 - 2
*(a^4*x^4 - a^2*x^2)*log(a*x + 1)*log(-a*x + 1) + (a^4*x^4 - a^2*x^2)*log(-a*x + 1)^2) - integrate(-2*(3*a^4*x
^4 - 2*a^2*x^2 + 1)/((a^6*x^7 - 2*a^4*x^5 + a^2*x^3)*log(a*x + 1) - (a^6*x^7 - 2*a^4*x^5 + a^2*x^3)*log(-a*x +
1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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