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

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

[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

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Rubi [A]  time = 0.0616321, antiderivative size = 82, 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} \[ -\frac{2 b^2}{(1-a)^3 x}+\frac{2 b^3 \log (x)}{(1-a)^4}-\frac{2 b^3 \log (-a-b x+1)}{(1-a)^4}-\frac{b}{(1-a)^2 x^2}-\frac{a+1}{3 (1-a) x^3} \]

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 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^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 = 0.0469672, 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)

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Maple [A]  time = 0.038, size = 76, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( 3\,a-3 \right ){x}^{3}}}+{\frac{a}{ \left ( 3\,a-3 \right ){x}^{3}}}-{\frac{b}{ \left ( a-1 \right ) ^{2}{x}^{2}}}+2\,{\frac{{b}^{3}\ln \left ( x \right ) }{ \left ( a-1 \right ) ^{4}}}+2\,{\frac{{b}^{2}}{ \left ( a-1 \right ) ^{3}x}}-2\,{\frac{{b}^{3}\ln \left ( bx+a-1 \right ) }{ \left ( a-1 \right ) ^{4}}} \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^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)

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Maxima [A]  time = 0.956396, size = 146, normalized size = 1.78 \begin{align*} -\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 \left (x\right )}{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}} \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^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)

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Fricas [A]  time = 1.87461, size = 216, normalized size = 2.63 \begin{align*} -\frac{6 \, b^{3} x^{3} \log \left (b x + a - 1\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 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}} \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^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)

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Sympy [B]  time = 0.79484, size = 260, normalized size = 3.17 \begin{align*} \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 )} \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**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))

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Giac [A]  time = 1.18719, size = 162, normalized size = 1.98 \begin{align*} -\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}} \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^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)