3.2.89 \(\int \frac {1-a^2 x^2}{\tanh ^{-1}(a x)^2} \, dx\) [189]

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

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 = 0.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-a^2 x^2}{\tanh ^{-1}(a x)^2} \, dx \end {gather*}

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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Maple [A]
time = 34.07, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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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((-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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

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

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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