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

Optimal. Leaf size=73 \[ \frac{9}{a c^3 (1-a x)}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{2}{3 a c^3 (1-a x)^3}+\frac{5 \log (1-a x)}{a c^3}+\frac{x}{c^3} \]

[Out]

x/c^3 + 2/(3*a*c^3*(1 - a*x)^3) - 7/(2*a*c^3*(1 - a*x)^2) + 9/(a*c^3*(1 - a*x)) + (5*Log[1 - a*x])/(a*c^3)

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Rubi [A]  time = 0.164493, antiderivative size = 73, 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} \[ \frac{9}{a c^3 (1-a x)}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{2}{3 a c^3 (1-a x)^3}+\frac{5 \log (1-a x)}{a c^3}+\frac{x}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^3 + 2/(3*a*c^3*(1 - a*x)^3) - 7/(2*a*c^3*(1 - a*x)^2) + 9/(a*c^3*(1 - a*x)) + (5*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 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 )^3} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx\\ &=\frac{a^3 \int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=\frac{a^3 \int \frac{x^3 (1+a x)}{(1-a x)^4} \, dx}{c^3}\\ &=\frac{a^3 \int \left (\frac{1}{a^3}+\frac{2}{a^3 (-1+a x)^4}+\frac{7}{a^3 (-1+a x)^3}+\frac{9}{a^3 (-1+a x)^2}+\frac{5}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=\frac{x}{c^3}+\frac{2}{3 a c^3 (1-a x)^3}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{9}{a c^3 (1-a x)}+\frac{5 \log (1-a x)}{a c^3}\\ \end{align*}

Mathematica [A]  time = 0.120448, size = 63, normalized size = 0.86 \[ \frac{6 a^4 x^4-18 a^3 x^3-36 a^2 x^2+81 a x+30 (a x-1)^3 \log (1-a x)-37}{6 a c^3 (a x-1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-37 + 81*a*x - 36*a^2*x^2 - 18*a^3*x^3 + 6*a^4*x^4 + 30*(-1 + a*x)^3*Log[1 - a*x])/(6*a*c^3*(-1 + a*x)^3)

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Maple [A]  time = 0.043, size = 66, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{3}}}-{\frac{7}{2\,a{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{2}{3\,a{c}^{3} \left ( ax-1 \right ) ^{3}}}+5\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{3}}}-9\,{\frac{1}{a{c}^{3} \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)^3,x)

[Out]

x/c^3-7/2/a/c^3/(a*x-1)^2-2/3/a/c^3/(a*x-1)^3+5/a/c^3*ln(a*x-1)-9/a/c^3/(a*x-1)

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Maxima [A]  time = 1.04371, size = 101, normalized size = 1.38 \begin{align*} -\frac{54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} + \frac{x}{c^{3}} + \frac{5 \, \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)*(a*x+1)/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

-1/6*(54*a^2*x^2 - 87*a*x + 37)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3) + x/c^3 + 5*log(a*x - 1)/(
a*c^3)

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Fricas [A]  time = 1.50469, size = 217, normalized size = 2.97 \begin{align*} \frac{6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 37}{6 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, 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)*(a*x+1)/(c-c/a/x)^3,x, algorithm="fricas")

[Out]

1/6*(6*a^4*x^4 - 18*a^3*x^3 - 36*a^2*x^2 + 81*a*x + 30*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(a*x - 1) - 37)/(a
^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)

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Sympy [A]  time = 0.534018, size = 73, normalized size = 1. \begin{align*} - \frac{54 a^{2} x^{2} - 87 a x + 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} + \frac{x}{c^{3}} + \frac{5 \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)*(a*x+1)/(c-c/a/x)**3,x)

[Out]

-(54*a**2*x**2 - 87*a*x + 37)/(6*a**4*c**3*x**3 - 18*a**3*c**3*x**2 + 18*a**2*c**3*x - 6*a*c**3) + x/c**3 + 5*
log(a*x - 1)/(a*c**3)

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Giac [A]  time = 1.12201, size = 68, normalized size = 0.93 \begin{align*} \frac{x}{c^{3}} + \frac{5 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{3}} - \frac{54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \,{\left (a x - 1\right )}^{3} a c^{3}} \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)^3,x, algorithm="giac")

[Out]

x/c^3 + 5*log(abs(a*x - 1))/(a*c^3) - 1/6*(54*a^2*x^2 - 87*a*x + 37)/((a*x - 1)^3*a*c^3)