### 3.395 $$\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^4} \, dx$$

Optimal. Leaf size=87 $\frac{14}{a c^4 (1-a x)}-\frac{8}{a c^4 (1-a x)^2}+\frac{3}{a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{6 \log (1-a x)}{a c^4}+\frac{x}{c^4}$

[Out]

x/c^4 - 1/(2*a*c^4*(1 - a*x)^4) + 3/(a*c^4*(1 - a*x)^3) - 8/(a*c^4*(1 - a*x)^2) + 14/(a*c^4*(1 - a*x)) + (6*Lo
g[1 - a*x])/(a*c^4)

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Rubi [A]  time = 0.172946, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {6167, 6131, 6129, 77} $\frac{14}{a c^4 (1-a x)}-\frac{8}{a c^4 (1-a x)^2}+\frac{3}{a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{6 \log (1-a x)}{a c^4}+\frac{x}{c^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])/(c - c/(a*x))^4,x]

[Out]

x/c^4 - 1/(2*a*c^4*(1 - a*x)^4) + 3/(a*c^4*(1 - a*x)^3) - 8/(a*c^4*(1 - a*x)^2) + 14/(a*c^4*(1 - a*x)) + (6*Lo
g[1 - a*x])/(a*c^4)

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 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 \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx\\ &=-\frac{a^4 \int \frac{e^{2 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=-\frac{a^4 \int \frac{x^4 (1+a x)}{(1-a x)^5} \, dx}{c^4}\\ &=-\frac{a^4 \int \left (-\frac{1}{a^4}-\frac{2}{a^4 (-1+a x)^5}-\frac{9}{a^4 (-1+a x)^4}-\frac{16}{a^4 (-1+a x)^3}-\frac{14}{a^4 (-1+a x)^2}-\frac{6}{a^4 (-1+a x)}\right ) \, dx}{c^4}\\ &=\frac{x}{c^4}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{3}{a c^4 (1-a x)^3}-\frac{8}{a c^4 (1-a x)^2}+\frac{14}{a c^4 (1-a x)}+\frac{6 \log (1-a x)}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.149446, size = 71, normalized size = 0.82 $\frac{2 a^5 x^5-8 a^4 x^4-16 a^3 x^3+60 a^2 x^2-56 a x+12 (a x-1)^4 \log (1-a x)+17}{2 a c^4 (a x-1)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^4,x]

[Out]

(17 - 56*a*x + 60*a^2*x^2 - 16*a^3*x^3 - 8*a^4*x^4 + 2*a^5*x^5 + 12*(-1 + a*x)^4*Log[1 - a*x])/(2*a*c^4*(-1 +
a*x)^4)

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Maple [A]  time = 0.044, size = 81, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{4}}}-8\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{2}}}-3\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{3}}}+6\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{4}}}-{\frac{1}{2\,a{c}^{4} \left ( ax-1 \right ) ^{4}}}-14\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(a*x-1)/(c-c/a/x)^4,x)

[Out]

x/c^4-8/a/c^4/(a*x-1)^2-3/a/c^4/(a*x-1)^3+6/a/c^4*ln(a*x-1)-1/2/a/c^4/(a*x-1)^4-14/a/c^4/(a*x-1)

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Maxima [A]  time = 1.02608, size = 126, normalized size = 1.45 \begin{align*} -\frac{28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac{x}{c^{4}} + \frac{6 \, \log \left (a x - 1\right )}{a c^{4}} \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)^4,x, algorithm="maxima")

[Out]

-1/2*(28*a^3*x^3 - 68*a^2*x^2 + 58*a*x - 17)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^
4) + x/c^4 + 6*log(a*x - 1)/(a*c^4)

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Fricas [A]  time = 1.55305, size = 271, normalized size = 3.11 \begin{align*} \frac{2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 56 \, a x + 12 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 17}{2 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \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)^4,x, algorithm="fricas")

[Out]

1/2*(2*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 60*a^2*x^2 - 56*a*x + 12*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x +
1)*log(a*x - 1) + 17)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)

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Sympy [A]  time = 0.648879, size = 94, normalized size = 1.08 \begin{align*} - \frac{28 a^{3} x^{3} - 68 a^{2} x^{2} + 58 a x - 17}{2 a^{5} c^{4} x^{4} - 8 a^{4} c^{4} x^{3} + 12 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 2 a c^{4}} + \frac{x}{c^{4}} + \frac{6 \log{\left (a x - 1 \right )}}{a c^{4}} \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)**4,x)

[Out]

-(28*a**3*x**3 - 68*a**2*x**2 + 58*a*x - 17)/(2*a**5*c**4*x**4 - 8*a**4*c**4*x**3 + 12*a**3*c**4*x**2 - 8*a**2
*c**4*x + 2*a*c**4) + x/c**4 + 6*log(a*x - 1)/(a*c**4)

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Giac [A]  time = 1.12609, size = 78, normalized size = 0.9 \begin{align*} \frac{x}{c^{4}} + \frac{6 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{4}} - \frac{28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \,{\left (a x - 1\right )}^{4} a c^{4}} \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)^4,x, algorithm="giac")

[Out]

x/c^4 + 6*log(abs(a*x - 1))/(a*c^4) - 1/2*(28*a^3*x^3 - 68*a^2*x^2 + 58*a*x - 17)/((a*x - 1)^4*a*c^4)