3.395 \(\int \frac {x^m \tanh ^{-1}(a x)^2}{(1-a^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {x^m \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {x^m \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{m} \operatorname {artanh}\left (a x\right )^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*x^2 + 1)*x^m*arctanh(a*x)^2/(a^4*x^4 - 2*a^2*x^2 + 1), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \arctanh \left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m\,{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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