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

Optimal. Leaf size=146 \[ \frac {501}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (a x+1)}-\frac {67}{16 a c^4 (1-a x)^2}+\frac {83}{48 a c^4 (1-a x)^3}-\frac {7}{16 a c^4 (1-a x)^4}+\frac {1}{20 a c^4 (1-a x)^5}+\frac {261 \log (1-a x)}{64 a c^4}-\frac {5 \log (a x+1)}{64 a c^4}+\frac {x}{c^4} \]

[Out]

x/c^4+1/20/a/c^4/(-a*x+1)^5-7/16/a/c^4/(-a*x+1)^4+83/48/a/c^4/(-a*x+1)^3-67/16/a/c^4/(-a*x+1)^2+501/64/a/c^4/(
-a*x+1)-1/64/a/c^4/(a*x+1)+261/64*ln(-a*x+1)/a/c^4-5/64*ln(a*x+1)/a/c^4

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Rubi [A]  time = 0.19, antiderivative size = 146, 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 = {6157, 6150, 88} \[ \frac {501}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (a x+1)}-\frac {67}{16 a c^4 (1-a x)^2}+\frac {83}{48 a c^4 (1-a x)^3}-\frac {7}{16 a c^4 (1-a x)^4}+\frac {1}{20 a c^4 (1-a x)^5}+\frac {261 \log (1-a x)}{64 a c^4}-\frac {5 \log (a x+1)}{64 a c^4}+\frac {x}{c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^4 + 1/(20*a*c^4*(1 - a*x)^5) - 7/(16*a*c^4*(1 - a*x)^4) + 83/(48*a*c^4*(1 - a*x)^3) - 67/(16*a*c^4*(1 - a*
x)^2) + 501/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (261*Log[1 - a*x])/(64*a*c^4) - (5*Log[1 + a*x])/(
64*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 6150

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p
] || GtQ[c, 0])

Rule 6157

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 - a^2*x^
2)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=\frac {a^8 \int \frac {e^{4 \tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=\frac {a^8 \int \frac {x^8}{(1-a x)^6 (1+a x)^2} \, dx}{c^4}\\ &=\frac {a^8 \int \left (\frac {1}{a^8}+\frac {1}{4 a^8 (-1+a x)^6}+\frac {7}{4 a^8 (-1+a x)^5}+\frac {83}{16 a^8 (-1+a x)^4}+\frac {67}{8 a^8 (-1+a x)^3}+\frac {501}{64 a^8 (-1+a x)^2}+\frac {261}{64 a^8 (-1+a x)}+\frac {1}{64 a^8 (1+a x)^2}-\frac {5}{64 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}+\frac {1}{20 a c^4 (1-a x)^5}-\frac {7}{16 a c^4 (1-a x)^4}+\frac {83}{48 a c^4 (1-a x)^3}-\frac {67}{16 a c^4 (1-a x)^2}+\frac {501}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (1+a x)}+\frac {261 \log (1-a x)}{64 a c^4}-\frac {5 \log (1+a x)}{64 a c^4}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 98, normalized size = 0.67 \[ \frac {\frac {2 \left (480 a^7 x^7-1920 a^6 x^6-1365 a^5 x^5+9300 a^4 x^4-6800 a^3 x^3-4900 a^2 x^2+7541 a x-2384\right )}{(a x-1)^5 (a x+1)}+3915 \log (1-a x)-75 \log (a x+1)}{960 a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(-2384 + 7541*a*x - 4900*a^2*x^2 - 6800*a^3*x^3 + 9300*a^4*x^4 - 1365*a^5*x^5 - 1920*a^6*x^6 + 480*a^7*x^7
))/((-1 + a*x)^5*(1 + a*x)) + 3915*Log[1 - a*x] - 75*Log[1 + a*x])/(960*a*c^4)

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fricas [A]  time = 0.66, size = 207, normalized size = 1.42 \[ \frac {960 \, a^{7} x^{7} - 3840 \, a^{6} x^{6} - 2730 \, a^{5} x^{5} + 18600 \, a^{4} x^{4} - 13600 \, a^{3} x^{3} - 9800 \, a^{2} x^{2} + 15082 \, a x - 75 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x + 1\right ) + 3915 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x - 1\right ) - 4768}{960 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \]

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^2/x^2)^4,x, algorithm="fricas")

[Out]

1/960*(960*a^7*x^7 - 3840*a^6*x^6 - 2730*a^5*x^5 + 18600*a^4*x^4 - 13600*a^3*x^3 - 9800*a^2*x^2 + 15082*a*x -
75*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x + 1) + 3915*(a^6*x^6 - 4*a^5*x^5 + 5*a^4*
x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(a*x - 1) - 4768)/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2
+ 4*a^2*c^4*x - a*c^4)

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giac [A]  time = 0.18, size = 96, normalized size = 0.66 \[ \frac {x}{c^{4}} - \frac {5 \, \log \left ({\left | a x + 1 \right |}\right )}{64 \, a c^{4}} + \frac {261 \, \log \left ({\left | a x - 1 \right |}\right )}{64 \, a c^{4}} - \frac {3765 \, a^{5} x^{5} - 9300 \, a^{4} x^{4} + 4400 \, a^{3} x^{3} + 6820 \, a^{2} x^{2} - 8021 \, a x + 2384}{480 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{5} a c^{4}} \]

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^2/x^2)^4,x, algorithm="giac")

[Out]

x/c^4 - 5/64*log(abs(a*x + 1))/(a*c^4) + 261/64*log(abs(a*x - 1))/(a*c^4) - 1/480*(3765*a^5*x^5 - 9300*a^4*x^4
+ 4400*a^3*x^3 + 6820*a^2*x^2 - 8021*a*x + 2384)/((a*x + 1)*(a*x - 1)^5*a*c^4)

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maple [A]  time = 0.05, size = 125, normalized size = 0.86 \[ \frac {x}{c^{4}}-\frac {1}{20 a \,c^{4} \left (a x -1\right )^{5}}-\frac {7}{16 a \,c^{4} \left (a x -1\right )^{4}}-\frac {83}{48 a \,c^{4} \left (a x -1\right )^{3}}-\frac {67}{16 a \,c^{4} \left (a x -1\right )^{2}}-\frac {501}{64 a \,c^{4} \left (a x -1\right )}+\frac {261 \ln \left (a x -1\right )}{64 a \,c^{4}}-\frac {1}{64 a \,c^{4} \left (a x +1\right )}-\frac {5 \ln \left (a x +1\right )}{64 a \,c^{4}} \]

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^2/x^2)^4,x)

[Out]

x/c^4-1/20/a/c^4/(a*x-1)^5-7/16/a/c^4/(a*x-1)^4-83/48/a/c^4/(a*x-1)^3-67/16/a/c^4/(a*x-1)^2-501/64/a/c^4/(a*x-
1)+261/64/a/c^4*ln(a*x-1)-1/64/a/c^4/(a*x+1)-5/64*ln(a*x+1)/a/c^4

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maxima [A]  time = 0.32, size = 135, normalized size = 0.92 \[ -\frac {3765 \, a^{5} x^{5} - 9300 \, a^{4} x^{4} + 4400 \, a^{3} x^{3} + 6820 \, a^{2} x^{2} - 8021 \, a x + 2384}{480 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {5 \, \log \left (a x + 1\right )}{64 \, a c^{4}} + \frac {261 \, \log \left (a x - 1\right )}{64 \, a c^{4}} \]

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^2/x^2)^4,x, algorithm="maxima")

[Out]

-1/480*(3765*a^5*x^5 - 9300*a^4*x^4 + 4400*a^3*x^3 + 6820*a^2*x^2 - 8021*a*x + 2384)/(a^7*c^4*x^6 - 4*a^6*c^4*
x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4) + x/c^4 - 5/64*log(a*x + 1)/(a*c^4) + 261/64*log(a*
x - 1)/(a*c^4)

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mupad [B]  time = 0.94, size = 131, normalized size = 0.90 \[ \frac {\frac {341\,a\,x^2}{24}-\frac {8021\,x}{480}+\frac {149}{30\,a}+\frac {55\,a^2\,x^3}{6}-\frac {155\,a^3\,x^4}{8}+\frac {251\,a^4\,x^5}{32}}{-a^6\,c^4\,x^6+4\,a^5\,c^4\,x^5-5\,a^4\,c^4\,x^4+5\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4}+\frac {x}{c^4}+\frac {261\,\ln \left (a\,x-1\right )}{64\,a\,c^4}-\frac {5\,\ln \left (a\,x+1\right )}{64\,a\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((341*a*x^2)/24 - (8021*x)/480 + 149/(30*a) + (55*a^2*x^3)/6 - (155*a^3*x^4)/8 + (251*a^4*x^5)/32)/(c^4 + 5*a^
2*c^4*x^2 - 5*a^4*c^4*x^4 + 4*a^5*c^4*x^5 - a^6*c^4*x^6 - 4*a*c^4*x) + x/c^4 + (261*log(a*x - 1))/(64*a*c^4) -
(5*log(a*x + 1))/(64*a*c^4)

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sympy [A]  time = 0.90, size = 144, normalized size = 0.99 \[ a^{8} \left (\frac {- 3765 a^{5} x^{5} + 9300 a^{4} x^{4} - 4400 a^{3} x^{3} - 6820 a^{2} x^{2} + 8021 a x - 2384}{480 a^{15} c^{4} x^{6} - 1920 a^{14} c^{4} x^{5} + 2400 a^{13} c^{4} x^{4} - 2400 a^{11} c^{4} x^{2} + 1920 a^{10} c^{4} x - 480 a^{9} c^{4}} + \frac {x}{a^{8} c^{4}} + \frac {\frac {261 \log {\left (x - \frac {1}{a} \right )}}{64} - \frac {5 \log {\left (x + \frac {1}{a} \right )}}{64}}{a^{9} c^{4}}\right ) \]

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**2/x**2)**4,x)

[Out]

a**8*((-3765*a**5*x**5 + 9300*a**4*x**4 - 4400*a**3*x**3 - 6820*a**2*x**2 + 8021*a*x - 2384)/(480*a**15*c**4*x
**6 - 1920*a**14*c**4*x**5 + 2400*a**13*c**4*x**4 - 2400*a**11*c**4*x**2 + 1920*a**10*c**4*x - 480*a**9*c**4)
+ x/(a**8*c**4) + (261*log(x - 1/a)/64 - 5*log(x + 1/a)/64)/(a**9*c**4))

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