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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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 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]

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 = 0.01, size = 13, normalized size = 1.00 \[ 2 \log (a x+1)-\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + 2*Log[1 + a*x]

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 13, normalized size = 1.00 \[ 2 \, \log \left (a x + 1\right ) - \log \relax (x) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 15, normalized size = 1.15 \[ 2 \, \log \left ({\left | a x + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 14, normalized size = 1.08 \[ -\ln \relax (x )+2 \ln \left (a x +1\right ) \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 13, normalized size = 1.00 \[ 2 \, \log \left (a x + 1\right ) - \log \relax (x) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 14, normalized size = 1.08 \[ 2\,\ln \left (-3\,a\,x-3\right )-\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x - 1)/(x*(a*x + 1)),x)

[Out]

2*log(- 3*a*x - 3) - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 10, normalized size = 0.77 \[ - \log {\relax (x )} + 2 \log {\left (x + \frac {1}{a} \right )} \]

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)

________________________________________________________________________________________