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

Optimal. Leaf size=21 \[ 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, 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, 77} \[ 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 2*a*Log[x] - 2*a*Log[1 - a*x]

Rule 77

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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 2*a*Log[x] - 2*a*Log[1 - a*x]

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 23, normalized size = 1.10 \[ -\frac {2 \, a x \log \left (a x - 1\right ) - 2 \, a x \log \relax (x) + 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*x*log(a*x - 1) - 2*a*x*log(x) + 1)/x

________________________________________________________________________________________

giac [A]  time = 0.25, size = 22, normalized size = 1.05 \[ -2 \, a \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a \log \left ({\left | x \right |}\right ) - \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*log(abs(a*x - 1)) + 2*a*log(abs(x)) - 1/x

________________________________________________________________________________________

maple [A]  time = 0.03, size = 21, normalized size = 1.00 \[ -\frac {1}{x}+2 a \ln \relax (x )-2 a \ln \left (a x -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 20, normalized size = 0.95 \[ -2 \, a \log \left (a x - 1\right ) + 2 \, a \log \relax (x) - \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*log(a*x - 1) + 2*a*log(x) - 1/x

________________________________________________________________________________________

mupad [B]  time = 0.81, size = 16, normalized size = 0.76 \[ 4\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a*atanh(2*a*x - 1) - 1/x

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 17, normalized size = 0.81 \[ - 2 a \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*(-log(x) + log(x - 1/a)) - 1/x

________________________________________________________________________________________