3.833 \(\int \frac{e^{2 \tanh ^{-1}(a+b x)}}{x^3} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 b^2 \log (x)}{(1-a)^3}-\frac{2 b^2 \log (-a-b x+1)}{(1-a)^3}-\frac{2 b}{(1-a)^2 x}-\frac{a+1}{2 (1-a) x^2} \]

[Out]

-(1 + a)/(2*(1 - a)*x^2) - (2*b)/((1 - a)^2*x) + (2*b^2*Log[x])/(1 - a)^3 - (2*b^2*Log[1 - a - b*x])/(1 - a)^3

________________________________________________________________________________________

Rubi [A]  time = 0.050526, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 77} \[ \frac{2 b^2 \log (x)}{(1-a)^3}-\frac{2 b^2 \log (-a-b x+1)}{(1-a)^3}-\frac{2 b}{(1-a)^2 x}-\frac{a+1}{2 (1-a) x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a + b*x])/x^3,x]

[Out]

-(1 + a)/(2*(1 - a)*x^2) - (2*b)/((1 - a)^2*x) + (2*b^2*Log[x])/(1 - a)^3 - (2*b^2*Log[1 - a - b*x])/(1 - a)^3

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, 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]))))

Rubi steps

\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac{1+a+b x}{x^3 (1-a-b x)} \, dx\\ &=\int \left (\frac{-1-a}{(-1+a) x^3}+\frac{2 b}{(-1+a)^2 x^2}-\frac{2 b^2}{(-1+a)^3 x}+\frac{2 b^3}{(-1+a)^3 (-1+a+b x)}\right ) \, dx\\ &=-\frac{1+a}{2 (1-a) x^2}-\frac{2 b}{(1-a)^2 x}+\frac{2 b^2 \log (x)}{(1-a)^3}-\frac{2 b^2 \log (1-a-b x)}{(1-a)^3}\\ \end{align*}

Mathematica [A]  time = 0.0333783, size = 54, normalized size = 0.81 \[ \frac{(a-1) \left (a^2-4 b x-1\right )+4 b^2 x^2 \log (-a-b x+1)-4 b^2 x^2 \log (x)}{2 (a-1)^3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a + b*x])/x^3,x]

[Out]

((-1 + a)*(-1 + a^2 - 4*b*x) - 4*b^2*x^2*Log[x] + 4*b^2*x^2*Log[1 - a - b*x])/(2*(-1 + a)^3*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 63, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( 2\,a-2 \right ){x}^{2}}}+{\frac{a}{ \left ( 2\,a-2 \right ){x}^{2}}}-2\,{\frac{b}{ \left ( a-1 \right ) ^{2}x}}-2\,{\frac{{b}^{2}\ln \left ( x \right ) }{ \left ( a-1 \right ) ^{3}}}+2\,{\frac{{b}^{2}\ln \left ( bx+a-1 \right ) }{ \left ( a-1 \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a+1)^2/(1-(b*x+a)^2)/x^3,x)

[Out]

1/2/(a-1)/x^2+1/2/(a-1)/x^2*a-2*b/(a-1)^2/x-2/(a-1)^3*b^2*ln(x)+2/(a-1)^3*b^2*ln(b*x+a-1)

________________________________________________________________________________________

Maxima [A]  time = 0.958169, size = 100, normalized size = 1.49 \begin{align*} \frac{2 \, b^{2} \log \left (b x + a - 1\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} - \frac{2 \, b^{2} \log \left (x\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} + \frac{a^{2} - 4 \, b x - 1}{2 \,{\left (a^{2} - 2 \, a + 1\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x^3,x, algorithm="maxima")

[Out]

2*b^2*log(b*x + a - 1)/(a^3 - 3*a^2 + 3*a - 1) - 2*b^2*log(x)/(a^3 - 3*a^2 + 3*a - 1) + 1/2*(a^2 - 4*b*x - 1)/
((a^2 - 2*a + 1)*x^2)

________________________________________________________________________________________

Fricas [A]  time = 1.88922, size = 161, normalized size = 2.4 \begin{align*} \frac{4 \, b^{2} x^{2} \log \left (b x + a - 1\right ) - 4 \, b^{2} x^{2} \log \left (x\right ) + a^{3} - 4 \,{\left (a - 1\right )} b x - a^{2} - a + 1}{2 \,{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x^3,x, algorithm="fricas")

[Out]

1/2*(4*b^2*x^2*log(b*x + a - 1) - 4*b^2*x^2*log(x) + a^3 - 4*(a - 1)*b*x - a^2 - a + 1)/((a^3 - 3*a^2 + 3*a -
1)*x^2)

________________________________________________________________________________________

Sympy [B]  time = 0.669218, size = 209, normalized size = 3.12 \begin{align*} - \frac{2 b^{2} \log{\left (x + \frac{- \frac{2 a^{4} b^{2}}{\left (a - 1\right )^{3}} + \frac{8 a^{3} b^{2}}{\left (a - 1\right )^{3}} - \frac{12 a^{2} b^{2}}{\left (a - 1\right )^{3}} + 2 a b^{2} + \frac{8 a b^{2}}{\left (a - 1\right )^{3}} - 2 b^{2} - \frac{2 b^{2}}{\left (a - 1\right )^{3}}}{4 b^{3}} \right )}}{\left (a - 1\right )^{3}} + \frac{2 b^{2} \log{\left (x + \frac{\frac{2 a^{4} b^{2}}{\left (a - 1\right )^{3}} - \frac{8 a^{3} b^{2}}{\left (a - 1\right )^{3}} + \frac{12 a^{2} b^{2}}{\left (a - 1\right )^{3}} + 2 a b^{2} - \frac{8 a b^{2}}{\left (a - 1\right )^{3}} - 2 b^{2} + \frac{2 b^{2}}{\left (a - 1\right )^{3}}}{4 b^{3}} \right )}}{\left (a - 1\right )^{3}} - \frac{- a^{2} + 4 b x + 1}{x^{2} \left (2 a^{2} - 4 a + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)**2/(1-(b*x+a)**2)/x**3,x)

[Out]

-2*b**2*log(x + (-2*a**4*b**2/(a - 1)**3 + 8*a**3*b**2/(a - 1)**3 - 12*a**2*b**2/(a - 1)**3 + 2*a*b**2 + 8*a*b
**2/(a - 1)**3 - 2*b**2 - 2*b**2/(a - 1)**3)/(4*b**3))/(a - 1)**3 + 2*b**2*log(x + (2*a**4*b**2/(a - 1)**3 - 8
*a**3*b**2/(a - 1)**3 + 12*a**2*b**2/(a - 1)**3 + 2*a*b**2 - 8*a*b**2/(a - 1)**3 - 2*b**2 + 2*b**2/(a - 1)**3)
/(4*b**3))/(a - 1)**3 - (-a**2 + 4*b*x + 1)/(x**2*(2*a**2 - 4*a + 2))

________________________________________________________________________________________

Giac [A]  time = 1.1441, size = 123, normalized size = 1.84 \begin{align*} \frac{2 \, b^{3} \log \left ({\left | b x + a - 1 \right |}\right )}{a^{3} b - 3 \, a^{2} b + 3 \, a b - b} - \frac{2 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3} - 3 \, a^{2} + 3 \, a - 1} + \frac{a^{3} - a^{2} - 4 \,{\left (a b - b\right )} x - a + 1}{2 \,{\left (a - 1\right )}^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a+1)^2/(1-(b*x+a)^2)/x^3,x, algorithm="giac")

[Out]

2*b^3*log(abs(b*x + a - 1))/(a^3*b - 3*a^2*b + 3*a*b - b) - 2*b^2*log(abs(x))/(a^3 - 3*a^2 + 3*a - 1) + 1/2*(a
^3 - a^2 - 4*(a*b - b)*x - a + 1)/((a - 1)^3*x^2)