### 3.411 $$\int \frac{e^{4 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^3} \, dx$$

Optimal. Leaf size=89 $\frac{19}{a c^3 (1-a x)}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{7 \log (1-a x)}{a c^3}+\frac{x}{c^3}$

[Out]

x/c^3 - 1/(a*c^3*(1 - a*x)^4) + 16/(3*a*c^3*(1 - a*x)^3) - 25/(2*a*c^3*(1 - a*x)^2) + 19/(a*c^3*(1 - a*x)) + (
7*Log[1 - a*x])/(a*c^3)

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Rubi [A]  time = 0.172051, antiderivative size = 89, 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, 88} $\frac{19}{a c^3 (1-a x)}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{7 \log (1-a x)}{a c^3}+\frac{x}{c^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(4*ArcCoth[a*x])/(c - c/(a*x))^3,x]

[Out]

x/c^3 - 1/(a*c^3*(1 - a*x)^4) + 16/(3*a*c^3*(1 - a*x)^3) - 25/(2*a*c^3*(1 - a*x)^2) + 19/(a*c^3*(1 - a*x)) + (
7*Log[1 - a*x])/(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 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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx\\ &=-\frac{a^3 \int \frac{e^{4 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac{a^3 \int \frac{x^3 (1+a x)^2}{(1-a x)^5} \, dx}{c^3}\\ &=-\frac{a^3 \int \left (-\frac{1}{a^3}-\frac{4}{a^3 (-1+a x)^5}-\frac{16}{a^3 (-1+a x)^4}-\frac{25}{a^3 (-1+a x)^3}-\frac{19}{a^3 (-1+a x)^2}-\frac{7}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=\frac{x}{c^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{19}{a c^3 (1-a x)}+\frac{7 \log (1-a x)}{a c^3}\\ \end{align*}

Mathematica [A]  time = 0.139355, size = 71, normalized size = 0.8 $\frac{6 a^5 x^5-24 a^4 x^4-78 a^3 x^3+243 a^2 x^2-218 a x+42 (a x-1)^4 \log (1-a x)+65}{6 a c^3 (a x-1)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(4*ArcCoth[a*x])/(c - c/(a*x))^3,x]

[Out]

(65 - 218*a*x + 243*a^2*x^2 - 78*a^3*x^3 - 24*a^4*x^4 + 6*a^5*x^5 + 42*(-1 + a*x)^4*Log[1 - a*x])/(6*a*c^3*(-1
+ a*x)^4)

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Maple [A]  time = 0.046, size = 81, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{3}}}-{\frac{25}{2\,a{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{16}{3\,a{c}^{3} \left ( ax-1 \right ) ^{3}}}+7\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{3}}}-{\frac{1}{a{c}^{3} \left ( ax-1 \right ) ^{4}}}-19\,{\frac{1}{a{c}^{3} \left ( ax-1 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)^2*(a*x+1)^2/(c-c/a/x)^3,x)

[Out]

x/c^3-25/2/a/c^3/(a*x-1)^2-16/3/a/c^3/(a*x-1)^3+7/a/c^3*ln(a*x-1)-1/a/c^3/(a*x-1)^4-19/a/c^3/(a*x-1)

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Maxima [A]  time = 1.04203, size = 126, normalized size = 1.42 \begin{align*} -\frac{114 \, a^{3} x^{3} - 267 \, a^{2} x^{2} + 224 \, a x - 65}{6 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac{x}{c^{3}} + \frac{7 \, \log \left (a x - 1\right )}{a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

-1/6*(114*a^3*x^3 - 267*a^2*x^2 + 224*a*x - 65)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a
*c^3) + x/c^3 + 7*log(a*x - 1)/(a*c^3)

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Fricas [A]  time = 1.84076, size = 275, normalized size = 3.09 \begin{align*} \frac{6 \, a^{5} x^{5} - 24 \, a^{4} x^{4} - 78 \, a^{3} x^{3} + 243 \, a^{2} x^{2} - 218 \, a x + 42 \,{\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 ) + 65}{6 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2/(c-c/a/x)^3,x, algorithm="fricas")

[Out]

1/6*(6*a^5*x^5 - 24*a^4*x^4 - 78*a^3*x^3 + 243*a^2*x^2 - 218*a*x + 42*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x
+ 1)*log(a*x - 1) + 65)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)

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Sympy [A]  time = 0.663173, size = 94, normalized size = 1.06 \begin{align*} - \frac{114 a^{3} x^{3} - 267 a^{2} x^{2} + 224 a x - 65}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} + \frac{x}{c^{3}} + \frac{7 \log{\left (a x - 1 \right )}}{a c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2/(c-c/a/x)**3,x)

[Out]

-(114*a**3*x**3 - 267*a**2*x**2 + 224*a*x - 65)/(6*a**5*c**3*x**4 - 24*a**4*c**3*x**3 + 36*a**3*c**3*x**2 - 24
*a**2*c**3*x + 6*a*c**3) + x/c**3 + 7*log(a*x - 1)/(a*c**3)

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Giac [A]  time = 1.15071, size = 147, normalized size = 1.65 \begin{align*} \frac{a x - 1}{a c^{3}} - \frac{7 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c^{3}} - \frac{\frac{114 \, a^{7} c^{9}}{a x - 1} + \frac{75 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac{32 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac{6 \, a^{7} c^{9}}{{\left (a x - 1\right )}^{4}}}{6 \, a^{8} c^{12}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2/(c-c/a/x)^3,x, algorithm="giac")

[Out]

(a*x - 1)/(a*c^3) - 7*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/(a*c^3) - 1/6*(114*a^7*c^9/(a*x - 1) + 75*a^7*c^9
/(a*x - 1)^2 + 32*a^7*c^9/(a*x - 1)^3 + 6*a^7*c^9/(a*x - 1)^4)/(a^8*c^12)