3.3.57 \(\int \frac {1}{x (1-a^2 x^2) \tanh ^{-1}(a x)^3} \, dx\) [257]

Optimal. Leaf size=37 \[ -\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a} \]

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

-1/2*1/(a*x*ArcTanh[a*x]^2) - 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 ) \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 4.33, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right ) \arctanh \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{a^{2} x^{3} \operatorname {atanh}^{3}{\left (a x \right )} - x \operatorname {atanh}^{3}{\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)/atanh(a*x)**3,x)

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^3\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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