### 3.27 $$\int e^{4 \coth ^{-1}(a x)} x \, dx$$

Optimal. Leaf size=39 $\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2}$

[Out]

(4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*Log[1 - a*x])/a^2

________________________________________________________________________________________

Rubi [A]  time = 0.0413822, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {6167, 6126, 77} $\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(4*ArcCoth[a*x])*x,x]

[Out]

(4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*Log[1 - a*x])/a^2

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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]

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 e^{4 \coth ^{-1}(a x)} x \, dx &=\int e^{4 \tanh ^{-1}(a x)} x \, dx\\ &=\int \frac{x (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac{4}{a}+x+\frac{4}{a (-1+a x)^2}+\frac{8}{a (-1+a x)}\right ) \, dx\\ &=\frac{4 x}{a}+\frac{x^2}{2}+\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0283569, size = 39, normalized size = 1. $\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(4*ArcCoth[a*x])*x,x]

[Out]

(4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*Log[1 - a*x])/a^2

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 36, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+4\,{\frac{x}{a}}-4\,{\frac{1}{{a}^{2} \left ( ax-1 \right ) }}+8\,{\frac{\ln \left ( ax-1 \right ) }{{a}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2+4*x/a-4/a^2/(a*x-1)+8/a^2*ln(a*x-1)

________________________________________________________________________________________

Maxima [A]  time = 1.08211, size = 55, normalized size = 1.41 \begin{align*} \frac{a x^{2} + 8 \, x}{2 \, a} - \frac{4}{a^{3} x - a^{2}} + \frac{8 \, \log \left (a x - 1\right )}{a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.83994, size = 109, normalized size = 2.79 \begin{align*} \frac{a^{3} x^{3} + 7 \, a^{2} x^{2} - 8 \, a x + 16 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 8}{2 \,{\left (a^{3} x - a^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^3*x^3 + 7*a^2*x^2 - 8*a*x + 16*(a*x - 1)*log(a*x - 1) - 8)/(a^3*x - a^2)

________________________________________________________________________________________

Sympy [A]  time = 0.327189, size = 31, normalized size = 0.79 \begin{align*} \frac{x^{2}}{2} - \frac{4}{a^{3} x - a^{2}} + \frac{4 x}{a} + \frac{8 \log{\left (a x - 1 \right )}}{a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12868, size = 86, normalized size = 2.21 \begin{align*} \frac{\frac{{\left (a x - 1\right )}^{2}{\left (\frac{10}{a x - 1} + 1\right )}}{a} - \frac{16 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{8}{{\left (a x - 1\right )} a}}{2 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a*x - 1)^2*(10/(a*x - 1) + 1)/a - 16*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/a - 8/((a*x - 1)*a))/a