3.5.65 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^4} \, dx\) [465]

Optimal. Leaf size=88 \[ -\frac {x}{c^4}+\frac {1}{2 a c^4 (1-a x)^4}-\frac {3}{a c^4 (1-a x)^3}+\frac {8}{a c^4 (1-a x)^2}-\frac {14}{a c^4 (1-a x)}-\frac {6 \log (1-a x)}{a c^4} \]

[Out]

-x/c^4+1/2/a/c^4/(-a*x+1)^4-3/a/c^4/(-a*x+1)^3+8/a/c^4/(-a*x+1)^2-14/a/c^4/(-a*x+1)-6*ln(-a*x+1)/a/c^4

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Rubi [A]
time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6266, 6264, 78} \begin {gather*} -\frac {14}{a c^4 (1-a x)}+\frac {8}{a c^4 (1-a x)^2}-\frac {3}{a c^4 (1-a x)^3}+\frac {1}{2 a c^4 (1-a x)^4}-\frac {6 \log (1-a x)}{a c^4}-\frac {x}{c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/c^4) + 1/(2*a*c^4*(1 - a*x)^4) - 3/(a*c^4*(1 - a*x)^3) + 8/(a*c^4*(1 - a*x)^2) - 14/(a*c^4*(1 - a*x)) - (6
*Log[1 - a*x])/(a*c^4)

Rule 78

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 6264

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 6266

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]

Rubi steps

\begin {align*} \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)}{(1-a x)^5} \, dx}{c^4}\\ &=\frac {a^4 \int \left (-\frac {1}{a^4}-\frac {2}{a^4 (-1+a x)^5}-\frac {9}{a^4 (-1+a x)^4}-\frac {16}{a^4 (-1+a x)^3}-\frac {14}{a^4 (-1+a x)^2}-\frac {6}{a^4 (-1+a x)}\right ) \, dx}{c^4}\\ &=-\frac {x}{c^4}+\frac {1}{2 a c^4 (1-a x)^4}-\frac {3}{a c^4 (1-a x)^3}+\frac {8}{a c^4 (1-a x)^2}-\frac {14}{a c^4 (1-a x)}-\frac {6 \log (1-a x)}{a c^4}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 71, normalized size = 0.81 \begin {gather*} \frac {-17+56 a x-60 a^2 x^2+16 a^3 x^3+8 a^4 x^4-2 a^5 x^5-12 (-1+a x)^4 \log (1-a x)}{2 a c^4 (-1+a x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-17 + 56*a*x - 60*a^2*x^2 + 16*a^3*x^3 + 8*a^4*x^4 - 2*a^5*x^5 - 12*(-1 + a*x)^4*Log[1 - a*x])/(2*a*c^4*(-1 +
a*x)^4)

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Maple [A]
time = 0.76, size = 74, normalized size = 0.84

method	result	size
risch	\(-\frac {x}{c^{4}}+\frac {14 a^{2} c^{4} x^{3}-34 c^{4} a \,x^{2}+29 c^{4} x -\frac {17 c^{4}}{2 a}}{c^{8} \left (a x -1\right )^{4}}-\frac {6 \ln \left (a x -1\right )}{a \,c^{4}}\)	\(68\)
default	\(\frac {a^{4} \left (-\frac {x}{a^{4}}+\frac {1}{2 \left (a x -1\right )^{4} a^{5}}+\frac {14}{\left (a x -1\right ) a^{5}}+\frac {8}{\left (a x -1\right )^{2} a^{5}}+\frac {3}{\left (a x -1\right )^{3} a^{5}}-\frac {6 \ln \left (a x -1\right )}{a^{5}}\right )}{c^{4}}\)	\(74\)
norman	\(\frac {-\frac {a^{4} x^{5}}{c}-\frac {6 x}{c}+\frac {21 a \,x^{2}}{c}-\frac {26 a^{2} x^{3}}{c}+\frac {25 a^{3} x^{4}}{2 c}}{\left (a x -1\right )^{4} c^{3}}-\frac {6 \ln \left (a x -1\right )}{a \,c^{4}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^4,x,method=_RETURNVERBOSE)

[Out]

a^4/c^4*(-x/a^4+1/2/(a*x-1)^4/a^5+14/(a*x-1)/a^5+8/(a*x-1)^2/a^5+3/(a*x-1)^3/a^5-6/a^5*ln(a*x-1))

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Maxima [A]
time = 0.26, size = 94, normalized size = 1.07 \begin {gather*} \frac {28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} - \frac {x}{c^{4}} - \frac {6 \, \log \left (a x - 1\right )}{a c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="maxima")

[Out]

1/2*(28*a^3*x^3 - 68*a^2*x^2 + 58*a*x - 17)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4
) - x/c^4 - 6*log(a*x - 1)/(a*c^4)

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Fricas [A]
time = 0.36, size = 126, normalized size = 1.43 \begin {gather*} -\frac {2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 56 \, a x + 12 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 17}{2 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="fricas")

[Out]

-1/2*(2*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 60*a^2*x^2 - 56*a*x + 12*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x +
1)*log(a*x - 1) + 17)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)

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Sympy [A]
time = 0.26, size = 95, normalized size = 1.08 \begin {gather*} - \frac {- 28 a^{3} x^{3} + 68 a^{2} x^{2} - 58 a x + 17}{2 a^{5} c^{4} x^{4} - 8 a^{4} c^{4} x^{3} + 12 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 2 a c^{4}} - \frac {x}{c^{4}} - \frac {6 \log {\left (a x - 1 \right )}}{a c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**4,x)

[Out]

-(-28*a**3*x**3 + 68*a**2*x**2 - 58*a*x + 17)/(2*a**5*c**4*x**4 - 8*a**4*c**4*x**3 + 12*a**3*c**4*x**2 - 8*a**
2*c**4*x + 2*a*c**4) - x/c**4 - 6*log(a*x - 1)/(a*c**4)

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Giac [A]
time = 0.43, size = 59, normalized size = 0.67 \begin {gather*} -\frac {x}{c^{4}} - \frac {6 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{4}} + \frac {28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \, {\left (a x - 1\right )}^{4} a c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a/x)^4,x, algorithm="giac")

[Out]

-x/c^4 - 6*log(abs(a*x - 1))/(a*c^4) + 1/2*(28*a^3*x^3 - 68*a^2*x^2 + 58*a*x - 17)/((a*x - 1)^4*a*c^4)

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Mupad [B]
time = 0.85, size = 90, normalized size = 1.02 \begin {gather*} \frac {29\,x-34\,a\,x^2-\frac {17}{2\,a}+14\,a^2\,x^3}{a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+6\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4}-\frac {x}{c^4}-\frac {6\,\ln \left (a\,x-1\right )}{a\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x + 1)^2/((c - c/(a*x))^4*(a^2*x^2 - 1)),x)

[Out]

(29*x - 34*a*x^2 - 17/(2*a) + 14*a^2*x^3)/(c^4 + 6*a^2*c^4*x^2 - 4*a^3*c^4*x^3 + a^4*c^4*x^4 - 4*a*c^4*x) - x/
c^4 - (6*log(a*x - 1))/(a*c^4)

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