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

Optimal. Leaf size=12 \[ \log (x)-2 \log (1-a x) \]

[Out]

Log[x] - 2*Log[1 - a*x]

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Rubi [A]  time = 0.0253327, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 72} \[ \log (x)-2 \log (1-a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - 2*Log[1 - a*x]

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0065252, size = 12, normalized size = 1. \[ \log (x)-2 \log (1-a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - 2*Log[1 - a*x]

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Maple [A]  time = 0.033, size = 12, normalized size = 1. \begin{align*} \ln \left ( x \right ) -2\,\ln \left ( ax-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)-2*ln(a*x-1)

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Maxima [A]  time = 0.951546, size = 15, normalized size = 1.25 \begin{align*} -2 \, \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)^2/(-a^2*x^2+1)/x,x, algorithm="maxima")

[Out]

-2*log(a*x - 1) + log(x)

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Fricas [A]  time = 1.76841, size = 35, normalized size = 2.92 \begin{align*} -2 \, \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)^2/(-a^2*x^2+1)/x,x, algorithm="fricas")

[Out]

-2*log(a*x - 1) + log(x)

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Sympy [A]  time = 0.2531, size = 10, normalized size = 0.83 \begin{align*} \log{\left (x \right )} - 2 \log{\left (x - \frac{1}{a} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - 2*log(x - 1/a)

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Giac [A]  time = 1.20251, size = 18, normalized size = 1.5 \begin{align*} -2 \, \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)^2/(-a^2*x^2+1)/x,x, algorithm="giac")

[Out]

-2*log(abs(a*x - 1)) + log(abs(x))