3.3.91 \(\int \frac {1}{x (1-a^2 x^2)^2 \tanh ^{-1}(a x)^2} \, dx\) [291]

Optimal. Leaf size=62 \[ -\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text {Chi}\left (2 \tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)},x\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.24, 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 {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 3.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 12.14, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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