3.924 \(\int \frac{e^{\tanh ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\log (1-a x)}{a} \]

[Out]

-(Log[1 - a*x]/a)

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Rubi [A]  time = 0.0341307, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6140, 31} \[ -\frac{\log (1-a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Log[1 - a*x]/a)

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0047174, size = 12, normalized size = 1. \[ -\frac{\log (1-a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Log[1 - a*x]/a)

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Maple [A]  time = 0.026, size = 12, normalized size = 1. \begin{align*} -{\frac{\ln \left ( ax-1 \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*ln(a*x-1)

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Maxima [A]  time = 0.962576, size = 15, normalized size = 1.25 \begin{align*} -\frac{\log \left (a x - 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1)/a

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Fricas [A]  time = 1.68061, size = 23, normalized size = 1.92 \begin{align*} -\frac{\log \left (a x - 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1)/a

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Sympy [A]  time = 0.062884, size = 8, normalized size = 0.67 \begin{align*} - \frac{\log{\left (a x - 1 \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1)/a

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Giac [A]  time = 1.15186, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \left ({\left | a x - 1 \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(a*x - 1))/a