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

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Rubi [A]  time = 0.16, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 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]))))

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 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 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]

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*}

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Mathematica [A]  time = 0.13, 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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fricas [A]  time = 0.49, size = 100, normalized size = 1.37 \[ \frac {6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 37}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]

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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giac [A]  time = 0.14, size = 50, normalized size = 0.68 \[ \frac {x}{c^{3}} + \frac {5 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{3}} - \frac {54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \, {\left (a x - 1\right )}^{3} a c^{3}} \]

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)

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maple [A]  time = 0.04, size = 66, normalized size = 0.90 \[ \frac {x}{c^{3}}+\frac {5 \ln \left (a x -1\right )}{a \,c^{3}}-\frac {9}{a \,c^{3} \left (a x -1\right )}-\frac {2}{3 a \,c^{3} \left (a x -1\right )^{3}}-\frac {7}{2 a \,c^{3} \left (a x -1\right )^{2}} \]

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

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maxima [A]  time = 0.31, size = 75, normalized size = 1.03 \[ -\frac {54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} + \frac {x}{c^{3}} + \frac {5 \, \log \left (a x - 1\right )}{a c^{3}} \]

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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mupad [B]  time = 0.08, size = 71, normalized size = 0.97 \[ \frac {9\,a\,x^2-\frac {29\,x}{2}+\frac {37}{6\,a}}{-a^3\,c^3\,x^3+3\,a^2\,c^3\,x^2-3\,a\,c^3\,x+c^3}+\frac {x}{c^3}+\frac {5\,\ln \left (a\,x-1\right )}{a\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*a*x^2 - (29*x)/2 + 37/(6*a))/(c^3 + 3*a^2*c^3*x^2 - a^3*c^3*x^3 - 3*a*c^3*x) + x/c^3 + (5*log(a*x - 1))/(a*
c^3)

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sympy [A]  time = 0.33, size = 73, normalized size = 1.00 \[ \frac {- 54 a^{2} x^{2} + 87 a x - 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} + \frac {x}{c^{3}} + \frac {5 \log {\left (a x - 1 \right )}}{a c^{3}} \]

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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