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

Optimal. Leaf size=49 \[ -\text {Int}\left (\frac {1}{x \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)},x\right )+\frac {3}{4} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{8} \text {Shi}\left (4 \tanh ^{-1}(a x)\right ) \]

[Out]

3/4*Shi(2*arctanh(a*x))+1/8*Shi(4*arctanh(a*x))-Unintegrable(1/x/(a^2*x^2-1)/arctanh(a*x),x)

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )} \operatorname {artanh}\left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{6} x^{7} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}{\left (a x \right )} - x \operatorname {atanh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(a**6*x**7*atanh(a*x) - 3*a**4*x**5*atanh(a*x) + 3*a**2*x**3*atanh(a*x) - x*atanh(a*x)), x)

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