### 3.28 $$\int e^{4 \coth ^{-1}(a x)} \, dx$$

Optimal. Leaf size=27 $\frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x$

[Out]

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

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Rubi [A]  time = 0.0191366, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {6167, 6125, 43} $\frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 6125

Int[E^(ArcTanh[(a_.)*(x_)]*(n_)), x_Symbol] :> Int[(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, n}, x] &&
!IntegerQ[(n - 1)/2]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0185358, size = 26, normalized size = 0.96 $-\frac{4}{a (a x-1)}+\frac{4 \log (1-a x)}{a}+x$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 26, normalized size = 1. \begin{align*} x-4\,{\frac{1}{a \left ( ax-1 \right ) }}+4\,{\frac{\ln \left ( ax-1 \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.998486, size = 35, normalized size = 1.3 \begin{align*} x + \frac{4 \, \log \left (a x - 1\right )}{a} - \frac{4}{a^{2} x - 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, algorithm="maxima")

[Out]

x + 4*log(a*x - 1)/a - 4/(a^2*x - a)

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Fricas [A]  time = 1.71151, size = 81, normalized size = 3. \begin{align*} \frac{a^{2} x^{2} - a x + 4 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 4}{a^{2} x - 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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.319388, size = 19, normalized size = 0.7 \begin{align*} x - \frac{4}{a^{2} x - a} + \frac{4 \log{\left (a x - 1 \right )}}{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)

[Out]

x - 4/(a**2*x - a) + 4*log(a*x - 1)/a

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Giac [A]  time = 1.12444, size = 62, normalized size = 2.3 \begin{align*} \frac{a x - 1}{a} - \frac{4 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{4}{{\left (a x - 1\right )} 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, algorithm="giac")

[Out]

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