3.460 \(\int \frac {1}{x (-4+x^2)^4} \, dx\)

Optimal. Leaf size=58 \[ \frac {1}{128 \left (4-x^2\right )}+\frac {1}{64 \left (4-x^2\right )^2}+\frac {1}{24 \left (4-x^2\right )^3}-\frac {1}{512} \log \left (4-x^2\right )+\frac {\log (x)}{256} \]

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Rubi [A]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {266, 44} \[ \frac {1}{128 \left (4-x^2\right )}+\frac {1}{64 \left (4-x^2\right )^2}+\frac {1}{24 \left (4-x^2\right )^3}-\frac {1}{512} \log \left (4-x^2\right )+\frac {\log (x)}{256} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-4 + x^2)^4),x]

[Out]

1/(24*(4 - x^2)^3) + 1/(64*(4 - x^2)^2) + 1/(128*(4 - x^2)) + Log[x]/256 - Log[4 - x^2]/512

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (-4+x^2\right )^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-4+x)^4 x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{4 (-4+x)^4}-\frac {1}{16 (-4+x)^3}+\frac {1}{64 (-4+x)^2}-\frac {1}{256 (-4+x)}+\frac {1}{256 x}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{24 \left (4-x^2\right )^3}+\frac {1}{64 \left (4-x^2\right )^2}+\frac {1}{128 \left (4-x^2\right )}+\frac {\log (x)}{256}-\frac {1}{512} \log \left (4-x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.69 \[ \frac {-3 \log \left (4-x^2\right )-\frac {4 \left (3 x^4-30 x^2+88\right )}{\left (x^2-4\right )^3}+6 \log (x)}{1536} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-4 + x^2)^4),x]

[Out]

((-4*(88 - 30*x^2 + 3*x^4))/(-4 + x^2)^3 + 6*Log[x] - 3*Log[4 - x^2])/1536

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IntegrateAlgebraic [A]  time = 0.03, size = 40, normalized size = 0.69 \[ -\frac {1}{512} \log \left (x^2-4\right )+\frac {-3 x^4+30 x^2-88}{384 \left (x^2-4\right )^3}+\frac {\log (x)}{256} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(-4 + x^2)^4),x]

[Out]

(-88 + 30*x^2 - 3*x^4)/(384*(-4 + x^2)^3) + Log[x]/256 - Log[-4 + x^2]/512

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fricas [A]  time = 1.14, size = 73, normalized size = 1.26 \[ -\frac {12 \, x^{4} - 120 \, x^{2} + 3 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )} \log \left (x^{2} - 4\right ) - 6 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )} \log \relax (x) + 352}{1536 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-4)^4,x, algorithm="fricas")

[Out]

-1/1536*(12*x^4 - 120*x^2 + 3*(x^6 - 12*x^4 + 48*x^2 - 64)*log(x^2 - 4) - 6*(x^6 - 12*x^4 + 48*x^2 - 64)*log(x
) + 352)/(x^6 - 12*x^4 + 48*x^2 - 64)

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giac [A]  time = 0.61, size = 42, normalized size = 0.72 \[ \frac {11 \, x^{6} - 156 \, x^{4} + 768 \, x^{2} - 1408}{3072 \, {\left (x^{2} - 4\right )}^{3}} + \frac {1}{512} \, \log \left (x^{2}\right ) - \frac {1}{512} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-4)^4,x, algorithm="giac")

[Out]

1/3072*(11*x^6 - 156*x^4 + 768*x^2 - 1408)/(x^2 - 4)^3 + 1/512*log(x^2) - 1/512*log(abs(x^2 - 4))

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maple [A]  time = 0.35, size = 34, normalized size = 0.59

method	result	size
risch	\(\frac {-\frac {1}{128} x^{4}+\frac {5}{64} x^{2}-\frac {11}{48}}{\left (x^{2}-4\right )^{3}}+\frac {\ln \relax (x )}{256}-\frac {\ln \left (x^{2}-4\right )}{512}\)	\(34\)
norman	\(\frac {-\frac {1}{128} x^{4}+\frac {5}{64} x^{2}-\frac {11}{48}}{\left (x^{2}-4\right )^{3}}+\frac {\ln \relax (x )}{256}-\frac {\ln \left (-2+x \right )}{512}-\frac {\ln \left (2+x \right )}{512}\)	\(38\)
meijerg	\(\frac {x^{2} \left (\frac {11}{16} x^{4}-\frac {27}{4} x^{2}+18\right )}{12288 \left (1-\frac {x^{2}}{4}\right )^{3}}-\frac {\ln \left (1-\frac {x^{2}}{4}\right )}{512}+\frac {11}{3072}+\frac {\ln \relax (x )}{256}-\frac {\ln \relax (2)}{256}+\frac {i \pi }{512}\)	\(51\)
default	\(\frac {\ln \relax (x )}{256}-\frac {1}{1536 \left (-2+x \right )^{3}}+\frac {3}{2048 \left (-2+x \right )^{2}}-\frac {11}{4096 \left (-2+x \right )}-\frac {\ln \left (-2+x \right )}{512}+\frac {1}{1536 \left (2+x \right )^{3}}+\frac {3}{2048 \left (2+x \right )^{2}}+\frac {11}{4096 \left (2+x \right )}-\frac {\ln \left (2+x \right )}{512}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^2-4)^4,x,method=_RETURNVERBOSE)

[Out]

(-1/128*x^4+5/64*x^2-11/48)/(x^2-4)^3+1/256*ln(x)-1/512*ln(x^2-4)

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maxima [A]  time = 0.42, size = 46, normalized size = 0.79 \[ -\frac {3 \, x^{4} - 30 \, x^{2} + 88}{384 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )}} - \frac {1}{512} \, \log \left (x^{2} - 4\right ) + \frac {1}{512} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-4)^4,x, algorithm="maxima")

[Out]

-1/384*(3*x^4 - 30*x^2 + 88)/(x^6 - 12*x^4 + 48*x^2 - 64) - 1/512*log(x^2 - 4) + 1/512*log(x^2)

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mupad [B]  time = 0.08, size = 44, normalized size = 0.76 \[ \frac {\ln \relax (x)}{256}-\frac {\ln \left (x^2-4\right )}{512}-\frac {\frac {x^4}{128}-\frac {5\,x^2}{64}+\frac {11}{48}}{x^6-12\,x^4+48\,x^2-64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 - 4)^4),x)

[Out]

log(x)/256 - log(x^2 - 4)/512 - (x^4/128 - (5*x^2)/64 + 11/48)/(48*x^2 - 12*x^4 + x^6 - 64)

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sympy [A]  time = 0.16, size = 41, normalized size = 0.71 \[ \frac {- 3 x^{4} + 30 x^{2} - 88}{384 x^{6} - 4608 x^{4} + 18432 x^{2} - 24576} + \frac {\log {\relax (x )}}{256} - \frac {\log {\left (x^{2} - 4 \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**2-4)**4,x)

[Out]

(-3*x**4 + 30*x**2 - 88)/(384*x**6 - 4608*x**4 + 18432*x**2 - 24576) + log(x)/256 - log(x**2 - 4)/512

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