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

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

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Rubi [A]  time = 0.15, antiderivative size = 105, 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 = {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*ArcTanh[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 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]

Rubi steps

\begin {align*} \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*}

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Mathematica [A]  time = 0.17, 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*ArcTanh[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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fricas [A]  time = 0.43, size = 154, normalized size = 1.47 \[ \frac {15 \, a^{6} x^{6} - 75 \, a^{5} x^{5} - 240 \, a^{4} x^{4} + 1080 \, a^{3} x^{3} - 1480 \, a^{2} x^{2} + 890 \, a x + 120 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (a x - 1\right ) - 202}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+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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giac [A]  time = 0.21, size = 66, normalized size = 0.63 \[ \frac {x}{c^{4}} + \frac {8 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{4}} - \frac {390 \, a^{4} x^{4} - 1230 \, a^{3} x^{3} + 1555 \, a^{2} x^{2} - 905 \, a x + 202}{15 \, {\left (a x - 1\right )}^{5} a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 96, normalized size = 0.91 \[ \frac {x}{c^{4}}-\frac {26}{a \,c^{4} \left (a x -1\right )}+\frac {8 \ln \left (a x -1\right )}{a \,c^{4}}-\frac {22}{a \,c^{4} \left (a x -1\right )^{2}}-\frac {4}{5 a \,c^{4} \left (a x -1\right )^{5}}-\frac {5}{a \,c^{4} \left (a x -1\right )^{4}}-\frac {41}{3 a \,c^{4} \left (a x -1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.33, size = 113, normalized size = 1.08 \[ -\frac {390 \, a^{4} x^{4} - 1230 \, a^{3} x^{3} + 1555 \, a^{2} x^{2} - 905 \, a x + 202}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac {x}{c^{4}} + \frac {8 \, \log \left (a x - 1\right )}{a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+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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mupad [B]  time = 0.10, size = 109, normalized size = 1.04 \[ \frac {x}{c^4}+\frac {\frac {311\,a\,x^2}{3}-\frac {181\,x}{3}+\frac {202}{15\,a}-82\,a^2\,x^3+26\,a^3\,x^4}{-a^5\,c^4\,x^5+5\,a^4\,c^4\,x^4-10\,a^3\,c^4\,x^3+10\,a^2\,c^4\,x^2-5\,a\,c^4\,x+c^4}+\frac {8\,\ln \left (a\,x-1\right )}{a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^4 + ((311*a*x^2)/3 - (181*x)/3 + 202/(15*a) - 82*a^2*x^3 + 26*a^3*x^4)/(c^4 + 10*a^2*c^4*x^2 - 10*a^3*c^4*
x^3 + 5*a^4*c^4*x^4 - a^5*c^4*x^5 - 5*a*c^4*x) + (8*log(a*x - 1))/(a*c^4)

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sympy [A]  time = 0.55, size = 114, normalized size = 1.09 \[ \frac {- 390 a^{4} x^{4} + 1230 a^{3} x^{3} - 1555 a^{2} x^{2} + 905 a x - 202}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} + \frac {x}{c^{4}} + \frac {8 \log {\left (a x - 1 \right )}}{a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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