3.31 \(\int \frac {e^{4 \tanh ^{-1}(a x)}}{x^2} \, dx\)

Optimal. Leaf size=32 \[ \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \]

[Out]

-1/x+4*a/(-a*x+1)+4*a*ln(x)-4*a*ln(-a*x+1)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 32, 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, 88} \[ \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])/x^2,x]

[Out]

-x^(-1) + (4*a)/(1 - a*x) + 4*a*Log[x] - 4*a*Log[1 - a*x]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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^{4 \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {(1+a x)^2}{x^2 (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {4 a}{x}+\frac {4 a^2}{(-1+a x)^2}-\frac {4 a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+\frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 32, normalized size = 1.00 \[ \frac {4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])/x^2,x]

[Out]

-x^(-1) + (4*a)/(1 - a*x) + 4*a*Log[x] - 4*a*Log[1 - a*x]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 55, normalized size = 1.72 \[ -\frac {5 \, a x + 4 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \, {\left (a^{2} x^{2} - a x\right )} \log \relax (x) - 1}{a x^{2} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/x^2,x, algorithm="fricas")

[Out]

-(5*a*x + 4*(a^2*x^2 - a*x)*log(a*x - 1) - 4*(a^2*x^2 - a*x)*log(x) - 1)/(a*x^2 - x)

________________________________________________________________________________________

giac [A]  time = 1.09, size = 36, normalized size = 1.12 \[ -4 \, a \log \left ({\left | a x - 1 \right |}\right ) + 4 \, a \log \left ({\left | x \right |}\right ) - \frac {5 \, a x - 1}{a x^{2} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/x^2,x, algorithm="giac")

[Out]

-4*a*log(abs(a*x - 1)) + 4*a*log(abs(x)) - (5*a*x - 1)/(a*x^2 - x)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 31, normalized size = 0.97 \[ -\frac {1}{x}+4 a \ln \relax (x )-\frac {4 a}{a x -1}-4 a \ln \left (a x -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2/x^2,x)

[Out]

-1/x+4*a*ln(x)-4*a/(a*x-1)-4*a*ln(a*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 34, normalized size = 1.06 \[ -4 \, a \log \left (a x - 1\right ) + 4 \, a \log \relax (x) - \frac {5 \, a x - 1}{a x^{2} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/x^2,x, algorithm="maxima")

[Out]

-4*a*log(a*x - 1) + 4*a*log(x) - (5*a*x - 1)/(a*x^2 - x)

________________________________________________________________________________________

mupad [B]  time = 0.81, size = 28, normalized size = 0.88 \[ 8\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )+\frac {5\,a\,x-1}{x-a\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)^4/(x^2*(a^2*x^2 - 1)^2),x)

[Out]

8*a*atanh(2*a*x - 1) + (5*a*x - 1)/(x - a*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 26, normalized size = 0.81 \[ 4 a \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {- 5 a x + 1}{a x^{2} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2/x**2,x)

[Out]

4*a*(log(x) - log(x - 1/a)) + (-5*a*x + 1)/(a*x**2 - x)

________________________________________________________________________________________