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

Optimal. Leaf size=82 \[ \frac {2 b^3 \log (x)}{(1-a)^4}-\frac {2 b^3 \log (-a-b x+1)}{(1-a)^4}-\frac {2 b^2}{(1-a)^3 x}-\frac {b}{(1-a)^2 x^2}-\frac {a+1}{3 (1-a) x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 b^2}{(1-a)^3 x}+\frac {2 b^3 \log (x)}{(1-a)^4}-\frac {2 b^3 \log (-a-b x+1)}{(1-a)^4}-\frac {b}{(1-a)^2 x^2}-\frac {a+1}{3 (1-a) x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a + b*x])/x^4,x]

[Out]

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

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 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]

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {1+a+b x}{x^4 (1-a-b x)} \, dx\\ &=\int \left (\frac {-1-a}{(-1+a) x^4}+\frac {2 b}{(-1+a)^2 x^3}-\frac {2 b^2}{(-1+a)^3 x^2}+\frac {2 b^3}{(-1+a)^4 x}-\frac {2 b^4}{(-1+a)^4 (-1+a+b x)}\right ) \, dx\\ &=-\frac {1+a}{3 (1-a) x^3}-\frac {b}{(1-a)^2 x^2}-\frac {2 b^2}{(1-a)^3 x}+\frac {2 b^3 \log (x)}{(1-a)^4}-\frac {2 b^3 \log (1-a-b x)}{(1-a)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 75, normalized size = 0.91 \[ \frac {(a-1) \left (a^3-a^2-3 a b x-a+6 b^2 x^2+3 b x+1\right )-6 b^3 x^3 \log (-a-b x+1)+6 b^3 x^3 \log (x)}{3 (a-1)^4 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + a)*(1 - a - a^2 + a^3 + 3*b*x - 3*a*b*x + 6*b^2*x^2) + 6*b^3*x^3*Log[x] - 6*b^3*x^3*Log[1 - a - b*x])/(
3*(-1 + a)^4*x^3)

________________________________________________________________________________________

fricas [A]  time = 1.41, size = 88, normalized size = 1.07 \[ -\frac {6 \, b^{3} x^{3} \log \left (b x + a - 1\right ) - 6 \, b^{3} x^{3} \log \relax (x) - 6 \, {\left (a - 1\right )} b^{2} x^{2} - a^{4} + 2 \, a^{3} + 3 \, {\left (a^{2} - 2 \, a + 1\right )} b x - 2 \, a + 1}{3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*b^3*x^3*log(b*x + a - 1) - 6*b^3*x^3*log(x) - 6*(a - 1)*b^2*x^2 - a^4 + 2*a^3 + 3*(a^2 - 2*a + 1)*b*x
- 2*a + 1)/((a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*x^3)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 120, normalized size = 1.46 \[ -\frac {2 \, b^{4} \log \left ({\left | b x + a - 1 \right |}\right )}{a^{4} b - 4 \, a^{3} b + 6 \, a^{2} b - 4 \, a b + b} + \frac {2 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1} + \frac {a^{4} - 2 \, a^{3} + 6 \, {\left (a b^{2} - b^{2}\right )} x^{2} - 3 \, {\left (a^{2} b - 2 \, a b + b\right )} x + 2 \, a - 1}{3 \, {\left (a - 1\right )}^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^4*log(abs(b*x + a - 1))/(a^4*b - 4*a^3*b + 6*a^2*b - 4*a*b + b) + 2*b^3*log(abs(x))/(a^4 - 4*a^3 + 6*a^2
- 4*a + 1) + 1/3*(a^4 - 2*a^3 + 6*(a*b^2 - b^2)*x^2 - 3*(a^2*b - 2*a*b + b)*x + 2*a - 1)/((a - 1)^4*x^3)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 76, normalized size = 0.93 \[ \frac {1}{3 \left (a -1\right ) x^{3}}+\frac {a}{3 \left (a -1\right ) x^{3}}-\frac {b}{\left (a -1\right )^{2} x^{2}}+\frac {2 b^{3} \ln \relax (x )}{\left (a -1\right )^{4}}+\frac {2 b^{2}}{\left (a -1\right )^{3} x}-\frac {2 b^{3} \ln \left (b x +a -1\right )}{\left (a -1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 108, normalized size = 1.32 \[ -\frac {2 \, b^{3} \log \left (b x + a - 1\right )}{a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1} + \frac {2 \, b^{3} \log \relax (x)}{a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1} + \frac {6 \, b^{2} x^{2} + a^{3} - 3 \, {\left (a - 1\right )} b x - a^{2} - a + 1}{3 \, {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^3*log(b*x + a - 1)/(a^4 - 4*a^3 + 6*a^2 - 4*a + 1) + 2*b^3*log(x)/(a^4 - 4*a^3 + 6*a^2 - 4*a + 1) + 1/3*(
6*b^2*x^2 + a^3 - 3*(a - 1)*b*x - a^2 - a + 1)/((a^3 - 3*a^2 + 3*a - 1)*x^3)

________________________________________________________________________________________

mupad [B]  time = 0.90, size = 84, normalized size = 1.02 \[ \frac {\frac {a+1}{3\,\left (a-1\right )}+\frac {2\,b^2\,x^2}{{\left (a-1\right )}^3}-\frac {b\,x}{{\left (a-1\right )}^2}}{x^3}-\frac {4\,b^3\,\mathrm {atanh}\left (\frac {a^4-4\,a^3+6\,a^2-4\,a+1}{{\left (a-1\right )}^4}+\frac {2\,b\,x}{a-1}\right )}{{\left (a-1\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a + 1)/(3*(a - 1)) + (2*b^2*x^2)/(a - 1)^3 - (b*x)/(a - 1)^2)/x^3 - (4*b^3*atanh((6*a^2 - 4*a - 4*a^3 + a^4
+ 1)/(a - 1)^4 + (2*b*x)/(a - 1)))/(a - 1)^4

________________________________________________________________________________________

sympy [B]  time = 0.56, size = 260, normalized size = 3.17 \[ \frac {2 b^{3} \log {\left (x + \frac {- \frac {2 a^{5} b^{3}}{\left (a - 1\right )^{4}} + \frac {10 a^{4} b^{3}}{\left (a - 1\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a - 1\right )^{4}} + \frac {20 a^{2} b^{3}}{\left (a - 1\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a - 1\right )^{4}} - 2 b^{3} + \frac {2 b^{3}}{\left (a - 1\right )^{4}}}{4 b^{4}} \right )}}{\left (a - 1\right )^{4}} - \frac {2 b^{3} \log {\left (x + \frac {\frac {2 a^{5} b^{3}}{\left (a - 1\right )^{4}} - \frac {10 a^{4} b^{3}}{\left (a - 1\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a - 1\right )^{4}} - \frac {20 a^{2} b^{3}}{\left (a - 1\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a - 1\right )^{4}} - 2 b^{3} - \frac {2 b^{3}}{\left (a - 1\right )^{4}}}{4 b^{4}} \right )}}{\left (a - 1\right )^{4}} - \frac {- a^{3} + a^{2} + a - 6 b^{2} x^{2} + x \left (3 a b - 3 b\right ) - 1}{x^{3} \left (3 a^{3} - 9 a^{2} + 9 a - 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**3*log(x + (-2*a**5*b**3/(a - 1)**4 + 10*a**4*b**3/(a - 1)**4 - 20*a**3*b**3/(a - 1)**4 + 20*a**2*b**3/(a
- 1)**4 + 2*a*b**3 - 10*a*b**3/(a - 1)**4 - 2*b**3 + 2*b**3/(a - 1)**4)/(4*b**4))/(a - 1)**4 - 2*b**3*log(x +
(2*a**5*b**3/(a - 1)**4 - 10*a**4*b**3/(a - 1)**4 + 20*a**3*b**3/(a - 1)**4 - 20*a**2*b**3/(a - 1)**4 + 2*a*b*
*3 + 10*a*b**3/(a - 1)**4 - 2*b**3 - 2*b**3/(a - 1)**4)/(4*b**4))/(a - 1)**4 - (-a**3 + a**2 + a - 6*b**2*x**2
+ x*(3*a*b - 3*b) - 1)/(x**3*(3*a**3 - 9*a**2 + 9*a - 3))

________________________________________________________________________________________