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

Optimal. Leaf size=105 \[ \frac{26}{a c^4 (1-a x)}-\frac{22}{a c^4 (1-a x)^2}+\frac{41}{3 a c^4 (1-a x)^3}-\frac{5}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5}+\frac{8 \log (1-a x)}{a c^4}+\frac{x}{c^4} \]

[Out]

x/c^4 + 4/(5*a*c^4*(1 - a*x)^5) - 5/(a*c^4*(1 - a*x)^4) + 41/(3*a*c^4*(1 - a*x)^3) - 22/(a*c^4*(1 - a*x)^2) +
26/(a*c^4*(1 - a*x)) + (8*Log[1 - a*x])/(a*c^4)

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Rubi [A]  time = 0.18284, antiderivative size = 105, 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} \[ \frac{26}{a c^4 (1-a x)}-\frac{22}{a c^4 (1-a x)^2}+\frac{41}{3 a c^4 (1-a x)^3}-\frac{5}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5}+\frac{8 \log (1-a x)}{a c^4}+\frac{x}{c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^4 + 4/(5*a*c^4*(1 - a*x)^5) - 5/(a*c^4*(1 - a*x)^4) + 41/(3*a*c^4*(1 - a*x)^3) - 22/(a*c^4*(1 - a*x)^2) +
26/(a*c^4*(1 - a*x)) + (8*Log[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 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 )^4} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx\\ &=\frac{a^4 \int \frac{e^{4 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac{a^4 \int \frac{x^4 (1+a x)^2}{(1-a x)^6} \, dx}{c^4}\\ &=\frac{a^4 \int \left (\frac{1}{a^4}+\frac{4}{a^4 (-1+a x)^6}+\frac{20}{a^4 (-1+a x)^5}+\frac{41}{a^4 (-1+a x)^4}+\frac{44}{a^4 (-1+a x)^3}+\frac{26}{a^4 (-1+a x)^2}+\frac{8}{a^4 (-1+a x)}\right ) \, dx}{c^4}\\ &=\frac{x}{c^4}+\frac{4}{5 a c^4 (1-a x)^5}-\frac{5}{a c^4 (1-a x)^4}+\frac{41}{3 a c^4 (1-a x)^3}-\frac{22}{a c^4 (1-a x)^2}+\frac{26}{a c^4 (1-a x)}+\frac{8 \log (1-a x)}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.172816, size = 79, normalized size = 0.75 \[ \frac{15 a^6 x^6-75 a^5 x^5-240 a^4 x^4+1080 a^3 x^3-1480 a^2 x^2+890 a x+120 (a x-1)^5 \log (1-a x)-202}{15 a c^4 (a x-1)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-202 + 890*a*x - 1480*a^2*x^2 + 1080*a^3*x^3 - 240*a^4*x^4 - 75*a^5*x^5 + 15*a^6*x^6 + 120*(-1 + a*x)^5*Log[1
- a*x])/(15*a*c^4*(-1 + a*x)^5)

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Maple [A]  time = 0.047, size = 96, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{4}}}-22\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{41}{3\,a{c}^{4} \left ( ax-1 \right ) ^{3}}}-{\frac{4}{5\,a{c}^{4} \left ( ax-1 \right ) ^{5}}}+8\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{4}}}-5\,{\frac{1}{a{c}^{4} \left ( ax-1 \right ) ^{4}}}-26\,{\frac{1}{a{c}^{4} \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)^4,x)

[Out]

x/c^4-22/a/c^4/(a*x-1)^2-41/3/a/c^4/(a*x-1)^3-4/5/a/c^4/(a*x-1)^5+8/a/c^4*ln(a*x-1)-5/a/c^4/(a*x-1)^4-26/a/c^4
/(a*x-1)

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Maxima [A]  time = 1.01582, size = 153, normalized size = 1.46 \begin{align*} -\frac{390 \, a^{4} x^{4} - 1230 \, a^{3} x^{3} + 1555 \, a^{2} x^{2} - 905 \, a x + 202}{15 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac{x}{c^{4}} + \frac{8 \, \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)^2*(a*x+1)^2/(c-c/a/x)^4,x, algorithm="maxima")

[Out]

-1/15*(390*a^4*x^4 - 1230*a^3*x^3 + 1555*a^2*x^2 - 905*a*x + 202)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^
3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4) + x/c^4 + 8*log(a*x - 1)/(a*c^4)

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Fricas [A]  time = 1.82345, size = 347, normalized size = 3.3 \begin{align*} \frac{15 \, a^{6} x^{6} - 75 \, a^{5} x^{5} - 240 \, a^{4} x^{4} + 1080 \, a^{3} x^{3} - 1480 \, a^{2} x^{2} + 890 \, a x + 120 \,{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (a x - 1\right ) - 202}{15 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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)^2*(a*x+1)^2/(c-c/a/x)^4,x, algorithm="fricas")

[Out]

1/15*(15*a^6*x^6 - 75*a^5*x^5 - 240*a^4*x^4 + 1080*a^3*x^3 - 1480*a^2*x^2 + 890*a*x + 120*(a^5*x^5 - 5*a^4*x^4
+ 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(a*x - 1) - 202)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10
*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)

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Sympy [A]  time = 0.797477, size = 114, normalized size = 1.09 \begin{align*} - \frac{390 a^{4} x^{4} - 1230 a^{3} x^{3} + 1555 a^{2} x^{2} - 905 a x + 202}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} + \frac{x}{c^{4}} + \frac{8 \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)**2*(a*x+1)**2/(c-c/a/x)**4,x)

[Out]

-(390*a**4*x**4 - 1230*a**3*x**3 + 1555*a**2*x**2 - 905*a*x + 202)/(15*a**6*c**4*x**5 - 75*a**5*c**4*x**4 + 15
0*a**4*c**4*x**3 - 150*a**3*c**4*x**2 + 75*a**2*c**4*x - 15*a*c**4) + x/c**4 + 8*log(a*x - 1)/(a*c**4)

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Giac [A]  time = 1.15734, size = 167, normalized size = 1.59 \begin{align*} \frac{a x - 1}{a c^{4}} - \frac{8 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a c^{4}} - \frac{\frac{390 \, a^{9} c^{16}}{a x - 1} + \frac{330 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{2}} + \frac{205 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{3}} + \frac{75 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{4}} + \frac{12 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{5}}}{15 \, a^{10} c^{20}} \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)^4,x, algorithm="giac")

[Out]

(a*x - 1)/(a*c^4) - 8*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/(a*c^4) - 1/15*(390*a^9*c^16/(a*x - 1) + 330*a^9*
c^16/(a*x - 1)^2 + 205*a^9*c^16/(a*x - 1)^3 + 75*a^9*c^16/(a*x - 1)^4 + 12*a^9*c^16/(a*x - 1)^5)/(a^10*c^20)