\(\int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx\) [46]

Optimal result

Integrand size = 12, antiderivative size = 13 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=-\log (x)+2 \log (1+a x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6261, 78} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \log (a x+1)-\log (x) \]

[In]

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

[Out]

-Log[x] + 2*Log[1 + a*x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 6261

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 6302

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*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(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] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=-\log (x)+2 \log (1+a x) \]

[In]

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

[Out]

-Log[x] + 2*Log[1 + a*x]

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
default	\(-\ln \left (x \right )+2 \ln \left (a x +1\right )\)	\(14\)
norman	\(-\ln \left (x \right )+2 \ln \left (a x +1\right )\)	\(14\)
parallelrisch	\(-\ln \left (x \right )+2 \ln \left (a x +1\right )\)	\(14\)
risch	\(2 \ln \left (-a x -1\right )-\ln \left (x \right )\)	\(15\)
meijerg	\(2 \ln \left (a x +1\right )-\ln \left (x \right )-\ln \left (a \right )\)	\(18\)

[In]

int((a*x-1)/(a*x+1)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \, \log \left (a x + 1\right ) - \log \left (x\right ) \]

[In]

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

[Out]

2*log(a*x + 1) - log(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=- \log {\left (x \right )} + 2 \log {\left (x + \frac {1}{a} \right )} \]

[In]

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

[Out]

-log(x) + 2*log(x + 1/a)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \, \log \left (a x + 1\right ) - \log \left (x\right ) \]

[In]

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

[Out]

2*log(a*x + 1) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2 \, \log \left ({\left | a x + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

2*log(abs(a*x + 1)) - log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{x} \, dx=2\,\ln \left (-3\,a\,x-3\right )-\ln \left (x\right ) \]

[In]

int((a*x - 1)/(x*(a*x + 1)),x)

[Out]

2*log(- 3*a*x - 3) - log(x)