3.4.87 \(\int \frac {x^m \tanh ^{-1}(a x)}{(1-a^2 x^2)^{3/2}} \, dx\) [387]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, 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 {x^m \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 4.18, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \arctanh \left (a x \right )}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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