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

Optimal. Leaf size=71 \[ \frac{13}{a c^2 (1-a x)}-\frac{6}{a c^2 (1-a x)^2}+\frac{4}{3 a c^2 (1-a x)^3}+\frac{6 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]

[Out]

x/c^2 + 4/(3*a*c^2*(1 - a*x)^3) - 6/(a*c^2*(1 - a*x)^2) + 13/(a*c^2*(1 - a*x)) + (6*Log[1 - a*x])/(a*c^2)

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Rubi [A]  time = 0.158327, antiderivative size = 71, 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} \[ \frac{13}{a c^2 (1-a x)}-\frac{6}{a c^2 (1-a x)^2}+\frac{4}{3 a c^2 (1-a x)^3}+\frac{6 \log (1-a x)}{a c^2}+\frac{x}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^2 + 4/(3*a*c^2*(1 - a*x)^3) - 6/(a*c^2*(1 - a*x)^2) + 13/(a*c^2*(1 - a*x)) + (6*Log[1 - a*x])/(a*c^2)

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 )^2} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx\\ &=\frac{a^2 \int \frac{e^{4 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac{a^2 \int \frac{x^2 (1+a x)^2}{(1-a x)^4} \, dx}{c^2}\\ &=\frac{a^2 \int \left (\frac{1}{a^2}+\frac{4}{a^2 (-1+a x)^4}+\frac{12}{a^2 (-1+a x)^3}+\frac{13}{a^2 (-1+a x)^2}+\frac{6}{a^2 (-1+a x)}\right ) \, dx}{c^2}\\ &=\frac{x}{c^2}+\frac{4}{3 a c^2 (1-a x)^3}-\frac{6}{a c^2 (1-a x)^2}+\frac{13}{a c^2 (1-a x)}+\frac{6 \log (1-a x)}{a c^2}\\ \end{align*}

Mathematica [A]  time = 0.105849, size = 63, normalized size = 0.89 \[ \frac{3 a^4 x^4-9 a^3 x^3-30 a^2 x^2+57 a x+18 (a x-1)^3 \log (1-a x)-25}{3 a c^2 (a x-1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-25 + 57*a*x - 30*a^2*x^2 - 9*a^3*x^3 + 3*a^4*x^4 + 18*(-1 + a*x)^3*Log[1 - a*x])/(3*a*c^2*(-1 + a*x)^3)

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Maple [A]  time = 0.046, size = 66, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{2}}}-6\,{\frac{1}{a{c}^{2} \left ( ax-1 \right ) ^{2}}}-{\frac{4}{3\,a{c}^{2} \left ( ax-1 \right ) ^{3}}}+6\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{2}}}-13\,{\frac{1}{a{c}^{2} \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)^2,x)

[Out]

x/c^2-6/a/c^2/(a*x-1)^2-4/3/a/c^2/(a*x-1)^3+6/a/c^2*ln(a*x-1)-13/a/c^2/(a*x-1)

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Maxima [A]  time = 1.01575, size = 101, normalized size = 1.42 \begin{align*} -\frac{39 \, a^{2} x^{2} - 60 \, a x + 25}{3 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} + \frac{x}{c^{2}} + \frac{6 \, \log \left (a x - 1\right )}{a c^{2}} \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)^2,x, algorithm="maxima")

[Out]

-1/3*(39*a^2*x^2 - 60*a*x + 25)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2) + x/c^2 + 6*log(a*x - 1)/(
a*c^2)

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Fricas [A]  time = 1.71908, size = 216, normalized size = 3.04 \begin{align*} \frac{3 \, a^{4} x^{4} - 9 \, a^{3} x^{3} - 30 \, a^{2} x^{2} + 57 \, a x + 18 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 25}{3 \,{\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\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)^2,x, algorithm="fricas")

[Out]

1/3*(3*a^4*x^4 - 9*a^3*x^3 - 30*a^2*x^2 + 57*a*x + 18*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 25)/(a^
4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)

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Sympy [A]  time = 0.542358, size = 73, normalized size = 1.03 \begin{align*} - \frac{39 a^{2} x^{2} - 60 a x + 25}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} + \frac{x}{c^{2}} + \frac{6 \log{\left (a x - 1 \right )}}{a c^{2}} \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)**2,x)

[Out]

-(39*a**2*x**2 - 60*a*x + 25)/(3*a**4*c**2*x**3 - 9*a**3*c**2*x**2 + 9*a**2*c**2*x - 3*a*c**2) + x/c**2 + 6*lo
g(a*x - 1)/(a*c**2)

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Giac [A]  time = 1.1339, size = 127, normalized size = 1.79 \begin{align*} \frac{a x - 1}{a c^{2}} - \frac{6 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c^{2}} - \frac{\frac{39 \, a^{5} c^{4}}{a x - 1} + \frac{18 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{2}} + \frac{4 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{3}}}{3 \, a^{6} c^{6}} \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)^2,x, algorithm="giac")

[Out]

(a*x - 1)/(a*c^2) - 6*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/(a*c^2) - 1/3*(39*a^5*c^4/(a*x - 1) + 18*a^5*c^4/
(a*x - 1)^2 + 4*a^5*c^4/(a*x - 1)^3)/(a^6*c^6)