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

Optimal. Leaf size=13 \[ \frac{\tanh ^{-1}(a x)^4}{4 a} \]

[Out]

ArcTanh[a*x]^4/(4*a)

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Rubi [A]  time = 0.0234723, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5948} \[ \frac{\tanh ^{-1}(a x)^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^3/(1 - a^2*x^2),x]

[Out]

ArcTanh[a*x]^4/(4*a)

Rule 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx &=\frac{\tanh ^{-1}(a x)^4}{4 a}\\ \end{align*}

Mathematica [A]  time = 0.005276, size = 13, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)^4}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[a*x]^4/(4*a)

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Maple [A]  time = 0.021, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arctanh(a*x)^4/a

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Maxima [B]  time = 0.982012, size = 282, normalized size = 21.69 \begin{align*} \frac{1}{2} \,{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right )^{3} + \frac{1}{64} \, a{\left (\frac{8 \,{\left (\log \left (a x + 1\right )^{3} - 3 \, \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 3 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )^{2} - \log \left (a x - 1\right )^{3}\right )} \operatorname{artanh}\left (a x\right )}{a^{2}} - \frac{\log \left (a x + 1\right )^{4} - 4 \, \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) + 6 \, \log \left (a x + 1\right )^{2} \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (a x - 1\right )^{3} + \log \left (a x - 1\right )^{4}}{a^{2}}\right )} - \frac{3 \,{\left (\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2}\right )} \operatorname{artanh}\left (a x\right )^{2}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(a*x + 1)/a - log(a*x - 1)/a)*arctanh(a*x)^3 + 1/64*a*(8*(log(a*x + 1)^3 - 3*log(a*x + 1)^2*log(a*x -
1) + 3*log(a*x + 1)*log(a*x - 1)^2 - log(a*x - 1)^3)*arctanh(a*x)/a^2 - (log(a*x + 1)^4 - 4*log(a*x + 1)^3*log
(a*x - 1) + 6*log(a*x + 1)^2*log(a*x - 1)^2 - 4*log(a*x + 1)*log(a*x - 1)^3 + log(a*x - 1)^4)/a^2) - 3/8*(log(
a*x + 1)^2 - 2*log(a*x + 1)*log(a*x - 1) + log(a*x - 1)^2)*arctanh(a*x)^2/a

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Fricas [A]  time = 2.20808, size = 49, normalized size = 3.77 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{4}}{64 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*log(-(a*x + 1)/(a*x - 1))^4/a

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Sympy [A]  time = 1.12115, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{4}{\left (a x \right )}}{4 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((atanh(a*x)**4/(4*a), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.18429, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )^{4}}{64 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*log(-(a*x + 1)/(a*x - 1))^4/a