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

Optimal. Leaf size=14 \[ -\frac{\log (a x+1)}{a c} \]

[Out]

-(Log[1 + a*x]/(a*c))

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Rubi [A]  time = 0.0546414, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 31} \[ -\frac{\log (a x+1)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)),x]

[Out]

-(Log[1 + a*x]/(a*c))

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 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0074188, size = 14, normalized size = 1. \[ -\frac{\log (a x+1)}{a c} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)),x]

[Out]

-(Log[1 + a*x]/(a*c))

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Maple [A]  time = 0.039, size = 15, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01747, size = 19, normalized size = 1.36 \begin{align*} -\frac{\log \left (a x + 1\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x + 1)/(a*c)

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Fricas [A]  time = 1.43935, size = 28, normalized size = 2. \begin{align*} -\frac{\log \left (a x + 1\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*x + 1)/(a*c)

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Sympy [A]  time = 0.242826, size = 12, normalized size = 0.86 \begin{align*} - \frac{\log{\left (a c x + c \right )}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(a*c*x + c)/(a*c)

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Giac [A]  time = 1.1383, size = 20, normalized size = 1.43 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(a*x + 1))/(a*c)