3.4.35 \(\int \frac {1}{x (1-a^2 x^2)^3 \tanh ^{-1}(a x)^2} \, dx\) [335]

Optimal. Leaf size=99 \[ -\frac {1}{a x \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Chi}\left (4 \tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)},x\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.45, 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 )^3 \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-(1/(a*x*ArcTanh[a*x])) - (a*x)/((1 - a^2*x^2)^2*ArcTanh[a*x]) - (a*x)/((1 - a^2*x^2)*ArcTanh[a*x]) + (3*CoshI
ntegral[2*ArcTanh[a*x]])/2 + CoshIntegral[4*ArcTanh[a*x]]/2 - 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 )^3 \tanh ^{-1}(a x)^2} \, dx &=a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {a x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+a \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\left (3 a^3\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, 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 )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+3 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\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+\text {Subst}\left (\int \frac {\cosh ^4(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 )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+3 \text {Subst}\left (\int \left (-\frac {1}{8 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\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 \left (\frac {3}{8 x}+\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, 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 )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{8} \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac {3}{8} \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\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 )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{2} \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Chi}\left (4 \tanh ^{-1}(a x)\right )+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 )^2 \tanh ^{-1}(a x)}-\frac {a x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{2} \text {Chi}\left (4 \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 = 3.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 5.47, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-2/((a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) - (a^5*x^5 - 2*a^3*x^3 + a*x)*log(-a*x + 1)) + integrate(-2*(5*a^
2*x^2 - 1)/((a^7*x^8 - 3*a^5*x^6 + 3*a^3*x^4 - a*x^2)*log(a*x + 1) - (a^7*x^8 - 3*a^5*x^6 + 3*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)^3/arctanh(a*x)^2,x, algorithm="fricas")

[Out]

integral(-1/((a^6*x^7 - 3*a^4*x^5 + 3*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}{a^{6} x^{7} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{2}{\left (a x \right )} - x \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)**3/atanh(a*x)**2,x)

[Out]

-Integral(1/(a**6*x**7*atanh(a*x)**2 - 3*a**4*x**5*atanh(a*x)**2 + 3*a**2*x**3*atanh(a*x)**2 - x*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)^3/arctanh(a*x)^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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