### 3.391 $$\int e^{2 \coth ^{-1}(a x)} (c-\frac{c}{a x}) \, dx$$

Optimal. Leaf size=11 $\frac{c \log (x)}{a}+c x$

[Out]

c*x + (c*Log[x])/a

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Rubi [A]  time = 0.0746432, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {6167, 6131, 6129, 43} $\frac{c \log (x)}{a}+c x$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x)),x]

[Out]

c*x + (c*Log[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 6131

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 + (c*x)/d)
^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

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

Mathematica [A]  time = 0.036415, size = 11, normalized size = 1. $\frac{c \log (x)}{a}+c x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x)),x]

[Out]

c*x + (c*Log[x])/a

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Maple [A]  time = 0.039, size = 12, normalized size = 1.1 \begin{align*} cx+{\frac{c\ln \left ( x \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(a*x-1)*(c-c/a/x),x)

[Out]

c*x+c*ln(x)/a

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Maxima [A]  time = 1.04582, size = 15, normalized size = 1.36 \begin{align*} c x + \frac{c \log \left (x\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x + c*log(x)/a

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Fricas [A]  time = 1.40232, size = 30, normalized size = 2.73 \begin{align*} \frac{a c x + c \log \left (x\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*c*x + c*log(x))/a

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Sympy [A]  time = 0.091203, size = 10, normalized size = 0.91 \begin{align*} \frac{a c x + c \log{\left (x \right )}}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*c*x + c*log(x))/a

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Giac [A]  time = 1.11006, size = 16, normalized size = 1.45 \begin{align*} c x + \frac{c \log \left ({\left | x \right |}\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x + c*log(abs(x))/a