3.5.91 \(\int \frac {\sqrt {1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx\) [491]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\sqrt {1-a^2 x^2}}{\tanh ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 {\sqrt {1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^3, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx &=\int \frac {\sqrt {1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-a^2 x^2}}{\tanh ^{-1}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 5.94, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-a^{2} x^{2}+1}}{\arctanh \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**(1/2)/atanh(a*x)**3,x)

[Out]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {1-a^2\,x^2}}{{\mathrm {atanh}\left (a\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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