### 3.46 $$\int \frac{e^{-2 \coth ^{-1}(a x)}}{x} \, dx$$

Optimal. Leaf size=13 $2 \log (a x+1)-\log (x)$

[Out]

-Log[x] + 2*Log[1 + a*x]

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Rubi [A]  time = 0.0401803, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6167, 6126, 72} $2 \log (a x+1)-\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(2*ArcCoth[a*x])*x),x]

[Out]

-Log[x] + 2*Log[1 + a*x]

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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 \coth ^{-1}(a x)}}{x} \, dx &=-\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 = 0.0071152, size = 13, normalized size = 1. $2 \log (a x+1)-\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*x),x]

[Out]

-Log[x] + 2*Log[1 + a*x]

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Maple [A]  time = 0.043, size = 14, normalized size = 1.1 \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(1/(a*x+1)*(a*x-1)/x,x)

[Out]

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

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Maxima [A]  time = 1.01063, size = 18, normalized size = 1.38 \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)/(a*x+1)/x,x, algorithm="maxima")

[Out]

2*log(a*x + 1) - log(x)

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Fricas [A]  time = 1.75448, size = 34, normalized size = 2.62 \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)/(a*x+1)/x,x, algorithm="fricas")

[Out]

2*log(a*x + 1) - log(x)

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Sympy [A]  time = 0.135858, size = 10, normalized size = 0.77 \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)/(a*x+1)/x,x)

[Out]

-log(x) + 2*log(x + 1/a)

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Giac [A]  time = 1.1399, size = 20, normalized size = 1.54 \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)/(a*x+1)/x,x, algorithm="giac")

[Out]

2*log(abs(a*x + 1)) - log(abs(x))