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

Optimal. Leaf size=17 \[ \frac{\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \]

[Out]

ArcTanh[a*x]^(1 + p)/(a*(1 + p))

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Rubi [A]  time = 0.0309873, antiderivative size = 17, 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} \[ \frac{\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^p/(1 - a^2*x^2),x]

[Out]

ArcTanh[a*x]^(1 + p)/(a*(1 + p))

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)^p}{1-a^2 x^2} \, dx &=\frac{\tanh ^{-1}(a x)^{1+p}}{a (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0112008, size = 17, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[a*x]^(1 + p)/(a*(1 + p))

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Maple [A]  time = 0.024, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{1+p}}{a \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(a*x)^(1+p)/a/(1+p)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\operatorname{artanh}\left (a x\right )^{p}}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(arctanh(a*x)^p/(a^2*x^2 - 1), x)

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Fricas [B]  time = 2.15072, size = 212, normalized size = 12.47 \begin{align*} \frac{\cosh \left (p \log \left (\frac{1}{2} \, \log \left (-\frac{a x + 1}{a x - 1}\right )\right )\right ) \log \left (-\frac{a x + 1}{a x - 1}\right ) + \log \left (-\frac{a x + 1}{a x - 1}\right ) \sinh \left (p \log \left (\frac{1}{2} \, \log \left (-\frac{a x + 1}{a x - 1}\right )\right )\right )}{2 \,{\left (a p + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cosh(p*log(1/2*log(-(a*x + 1)/(a*x - 1))))*log(-(a*x + 1)/(a*x - 1)) + log(-(a*x + 1)/(a*x - 1))*sinh(p*l
og(1/2*log(-(a*x + 1)/(a*x - 1)))))/(a*p + a)

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Sympy [A]  time = 2.97073, size = 26, normalized size = 1.53 \begin{align*} \begin{cases} \frac{\begin{cases} \frac{\operatorname{atanh}^{p + 1}{\left (a x \right )}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (\operatorname{atanh}{\left (a x \right )} \right )} & \text{otherwise} \end{cases}}{a} & \text{for}\: a \neq 0 \\0^{p} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((Piecewise((atanh(a*x)**(p + 1)/(p + 1), Ne(p, -1)), (log(atanh(a*x)), True))/a, Ne(a, 0)), (0**p*x,
True))

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Giac [A]  time = 1.1937, size = 41, normalized size = 2.41 \begin{align*} \frac{\left (\frac{1}{2} \, \log \left (-\frac{a x + 1}{a x - 1}\right )\right )^{p + 1}}{a{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/2*log(-(a*x + 1)/(a*x - 1)))^(p + 1)/(a*(p + 1))