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

Optimal. Leaf size=75 \[ \frac {7}{4 a c^4 (1-a x)}-\frac {1}{4 a c^4 (1-a x)^2}+\frac {17 \log (1-a x)}{8 a c^4}-\frac {\log (a x+1)}{8 a c^4}+\frac {x}{c^4} \]

[Out]

x/c^4-1/4/a/c^4/(-a*x+1)^2+7/4/a/c^4/(-a*x+1)+17/8*ln(-a*x+1)/a/c^4-1/8*ln(a*x+1)/a/c^4

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Rubi [A]  time = 0.16, antiderivative size = 75, 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, 88} \[ \frac {7}{4 a c^4 (1-a x)}-\frac {1}{4 a c^4 (1-a x)^2}+\frac {17 \log (1-a x)}{8 a c^4}-\frac {\log (a x+1)}{8 a c^4}+\frac {x}{c^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^4),x]

[Out]

x/c^4 - 1/(4*a*c^4*(1 - a*x)^2) + 7/(4*a*c^4*(1 - a*x)) + (17*Log[1 - a*x])/(8*a*c^4) - Log[1 + a*x]/(8*a*c^4)

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]))

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 )^4} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\\ &=-\frac {a^4 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=-\frac {a^4 \int \frac {x^4}{(1-a x)^3 (1+a x)} \, dx}{c^4}\\ &=-\frac {a^4 \int \left (-\frac {1}{a^4}-\frac {1}{2 a^4 (-1+a x)^3}-\frac {7}{4 a^4 (-1+a x)^2}-\frac {17}{8 a^4 (-1+a x)}+\frac {1}{8 a^4 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}-\frac {1}{4 a c^4 (1-a x)^2}+\frac {7}{4 a c^4 (1-a x)}+\frac {17 \log (1-a x)}{8 a c^4}-\frac {\log (1+a x)}{8 a c^4}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 73, normalized size = 0.97 \[ -\frac {7}{4 a c^4 (a x-1)}-\frac {1}{4 a c^4 (a x-1)^2}+\frac {17 \log (1-a x)}{8 a c^4}-\frac {\log (a x+1)}{8 a c^4}+\frac {x}{c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^4),x]

[Out]

x/c^4 - 1/(4*a*c^4*(-1 + a*x)^2) - 7/(4*a*c^4*(-1 + a*x)) + (17*Log[1 - a*x])/(8*a*c^4) - Log[1 + a*x]/(8*a*c^
4)

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fricas [A]  time = 0.55, size = 93, normalized size = 1.24 \[ \frac {8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 17 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 12}{8 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(c-c/a/x)^4,x, algorithm="fricas")

[Out]

1/8*(8*a^3*x^3 - 16*a^2*x^2 - 6*a*x - (a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 17*(a^2*x^2 - 2*a*x + 1)*log(a*x -
1) + 12)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)

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giac [A]  time = 0.14, size = 57, normalized size = 0.76 \[ \frac {x}{c^{4}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{4}} + \frac {17 \, \log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{4}} - \frac {7 \, a x - 6}{4 \, {\left (a x - 1\right )}^{2} a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(c-c/a/x)^4,x, algorithm="giac")

[Out]

x/c^4 - 1/8*log(abs(a*x + 1))/(a*c^4) + 17/8*log(abs(a*x - 1))/(a*c^4) - 1/4*(7*a*x - 6)/((a*x - 1)^2*a*c^4)

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maple [A]  time = 0.04, size = 65, normalized size = 0.87 \[ \frac {x}{c^{4}}-\frac {1}{4 a \,c^{4} \left (a x -1\right )^{2}}-\frac {7}{4 a \,c^{4} \left (a x -1\right )}+\frac {17 \ln \left (a x -1\right )}{8 a \,c^{4}}-\frac {\ln \left (a x +1\right )}{8 a \,c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)*(a*x-1)/(c-c/a/x)^4,x)

[Out]

x/c^4-1/4/a/c^4/(a*x-1)^2-7/4/a/c^4/(a*x-1)+17/8/a/c^4*ln(a*x-1)-1/8*ln(a*x+1)/a/c^4

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maxima [A]  time = 0.31, size = 69, normalized size = 0.92 \[ -\frac {7 \, a x - 6}{4 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {\log \left (a x + 1\right )}{8 \, a c^{4}} + \frac {17 \, \log \left (a x - 1\right )}{8 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(c-c/a/x)^4,x, algorithm="maxima")

[Out]

-1/4*(7*a*x - 6)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4) + x/c^4 - 1/8*log(a*x + 1)/(a*c^4) + 17/8*log(a*x - 1)/(a
*c^4)

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mupad [B]  time = 0.10, size = 68, normalized size = 0.91 \[ \frac {x}{c^4}-\frac {\frac {7\,x}{4}-\frac {3}{2\,a}}{a^2\,c^4\,x^2-2\,a\,c^4\,x+c^4}+\frac {17\,\ln \left (a\,x-1\right )}{8\,a\,c^4}-\frac {\ln \left (a\,x+1\right )}{8\,a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x - 1)/((c - c/(a*x))^4*(a*x + 1)),x)

[Out]

x/c^4 - ((7*x)/4 - 3/(2*a))/(c^4 + a^2*c^4*x^2 - 2*a*c^4*x) + (17*log(a*x - 1))/(8*a*c^4) - log(a*x + 1)/(8*a*
c^4)

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sympy [A]  time = 0.38, size = 73, normalized size = 0.97 \[ a^{4} \left (\frac {- 7 a x + 6}{4 a^{7} c^{4} x^{2} - 8 a^{6} c^{4} x + 4 a^{5} c^{4}} + \frac {x}{a^{4} c^{4}} + \frac {\frac {17 \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-1)/(a*x+1)/(c-c/a/x)**4,x)

[Out]

a**4*((-7*a*x + 6)/(4*a**7*c**4*x**2 - 8*a**6*c**4*x + 4*a**5*c**4) + x/(a**4*c**4) + (17*log(x - 1/a)/8 - log
(x + 1/a)/8)/(a**5*c**4))

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