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

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

[Out]

Log[x] - Log[1 - a*x]

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Rubi [A]  time = 0.0795024, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6150, 36, 29, 31} \[ \log (x)-\log (1-a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - Log[1 - a*x]

Rule 6150

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A]  time = 0.006461, size = 12, normalized size = 1. \[ \log (x)-\log (1-a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - Log[1 - a*x]

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Maple [A]  time = 0.032, size = 12, normalized size = 1. \begin{align*} \ln \left ( x \right ) -\ln \left ( ax-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.95007, size = 15, normalized size = 1.25 \begin{align*} -\log \left (a x - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1) + log(x)

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Fricas [A]  time = 1.76315, size = 32, normalized size = 2.67 \begin{align*} -\log \left (a x - 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x - 1) + log(x)

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Sympy [A]  time = 0.111181, size = 8, normalized size = 0.67 \begin{align*} \log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x - 1/a)

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Giac [A]  time = 1.15031, size = 18, normalized size = 1.5 \begin{align*} -\log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(a*x - 1)) + log(abs(x))