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

Optimal. Leaf size=20 \[ \text {Int}\left (\frac {1-a^2 x^2}{\tanh ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {1-a^2 x^2}{\tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - a^2*x^2)/ArcTanh[a*x]^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.40, size = 0, normalized size = 0.00 \[ \int \frac {1-a^2 x^2}{\tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} x^{2} - 1}{\operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )}}{a \log \left (a x + 1\right ) - a \log \left (-a x + 1\right )} - \int -\frac {8 \, {\left (a^{3} x^{3} - a x\right )}}{\log \left (a x + 1\right ) - \log \left (-a x + 1\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a^4*x^4 - 2*a^2*x^2 + 1)/(a*log(a*x + 1) - a*log(-a*x + 1)) - integrate(-8*(a^3*x^3 - a*x)/(log(a*x + 1) -
log(-a*x + 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ -\int \frac {a^2\,x^2-1}{{\mathrm {atanh}\left (a\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a^2*x^2 - 1)/atanh(a*x)^2,x)

[Out]

-int((a^2*x^2 - 1)/atanh(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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