### 3.213 $$\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx$$

Optimal. Leaf size=33 $-\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3}$

[Out]

-1/(2*a*c^3*(1 - a*x)) - ArcTanh[a*x]/(2*a*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0626256, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6167, 6129, 44, 207} $-\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^3),x]

[Out]

-1/(2*a*c^3*(1 - a*x)) - ArcTanh[a*x]/(2*a*c^3)

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^2 (1+a x)} \, dx}{c^3}\\ &=-\frac{\int \left (\frac{1}{2 (-1+a x)^2}-\frac{1}{2 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=-\frac{1}{2 a c^3 (1-a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{2 c^3}\\ &=-\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3}\\ \end{align*}

Mathematica [A]  time = 0.0199579, size = 32, normalized size = 0.97 $-\frac{\frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a}}{c^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^3),x]

[Out]

-((1/(2*a*(1 - a*x)) + ArcTanh[a*x]/(2*a))/c^3)

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 45, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{4\,a{c}^{3}}}+{\frac{1}{2\,a{c}^{3} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{4\,a{c}^{3}}} \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)^3,x)

[Out]

-1/4*ln(a*x+1)/a/c^3+1/2/c^3/a/(a*x-1)+1/4/c^3/a*ln(a*x-1)

________________________________________________________________________________________

Maxima [A]  time = 1.01879, size = 65, normalized size = 1.97 \begin{align*} \frac{1}{2 \,{\left (a^{2} c^{3} x - a c^{3}\right )}} - \frac{\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac{\log \left (a x - 1\right )}{4 \, a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a*c*x+c)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.52915, size = 108, normalized size = 3.27 \begin{align*} -\frac{{\left (a x - 1\right )} \log \left (a x + 1\right ) -{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (a^{2} c^{3} x - a c^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a*c*x+c)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.653654, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{2 a^{2} c^{3} x - 2 a c^{3}} - \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{4} + \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a*c*x+c)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1426, size = 62, normalized size = 1.88 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} + \frac{1}{2 \,{\left (a x - 1\right )} a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(-a*c*x+c)^3,x, algorithm="giac")

[Out]

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