3.76 \(\int \frac {1}{(3-19 x^2+32 x^4-16 x^6)^3} \, dx\)

Optimal. Leaf size=161 \[ \frac {5 x}{3 \left (3-4 x^2\right )}-\frac {2 x}{3 \left (3-4 x^2\right )^2}-\frac {7}{108 (1-2 x)}+\frac {67}{432 (1-x)}-\frac {67}{432 (x+1)}+\frac {7}{108 (2 x+1)}+\frac {1}{108 (1-2 x)^2}+\frac {1}{432 (1-x)^2}-\frac {1}{432 (x+1)^2}-\frac {1}{108 (2 x+1)^2}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )+\frac {5 \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{6 \sqrt {3}} \]

[Out]

1/108/(1-2*x)^2-7/108/(1-2*x)+1/432/(1-x)^2+67/432/(1-x)-1/432/(1+x)^2-67/432/(1+x)-1/108/(1+2*x)^2+7/108/(1+2
*x)-2/3*x/(-4*x^2+3)^2+5/3*x/(-4*x^2+3)+3913/648*arctanh(x)+67/162*arctanh(2*x)-67/18*arctanh(2/3*x*3^(1/2))*3
^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2057, 207, 199} \[ \frac {5 x}{3 \left (3-4 x^2\right )}-\frac {2 x}{3 \left (3-4 x^2\right )^2}-\frac {7}{108 (1-2 x)}+\frac {67}{432 (1-x)}-\frac {67}{432 (x+1)}+\frac {7}{108 (2 x+1)}+\frac {1}{108 (1-2 x)^2}+\frac {1}{432 (1-x)^2}-\frac {1}{432 (x+1)^2}-\frac {1}{108 (2 x+1)^2}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )+\frac {5 \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{6 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-3),x]

[Out]

1/(108*(1 - 2*x)^2) - 7/(108*(1 - 2*x)) + 1/(432*(1 - x)^2) + 67/(432*(1 - x)) - 1/(432*(1 + x)^2) - 67/(432*(
1 + x)) - 1/(108*(1 + 2*x)^2) + 7/(108*(1 + 2*x)) - (2*x)/(3*(3 - 4*x^2)^2) + (5*x)/(3*(3 - 4*x^2)) + (3913*Ar
cTanh[x])/648 + (67*ArcTanh[2*x])/162 + (5*ArcTanh[(2*x)/Sqrt[3]])/(6*Sqrt[3]) - 4*Sqrt[3]*ArcTanh[(2*x)/Sqrt[
3]]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx &=\int \left (-\frac {1}{216 (-1+x)^3}+\frac {67}{432 (-1+x)^2}+\frac {1}{216 (1+x)^3}+\frac {67}{432 (1+x)^2}-\frac {1}{27 (-1+2 x)^3}-\frac {7}{54 (-1+2 x)^2}+\frac {1}{27 (1+2 x)^3}-\frac {7}{54 (1+2 x)^2}-\frac {3913}{648 \left (-1+x^2\right )}+\frac {8}{\left (-3+4 x^2\right )^3}+\frac {12}{\left (-3+4 x^2\right )^2}+\frac {24}{-3+4 x^2}-\frac {67}{81 \left (-1+4 x^2\right )}\right ) \, dx\\ &=\frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {67}{81} \int \frac {1}{-1+4 x^2} \, dx-\frac {3913}{648} \int \frac {1}{-1+x^2} \, dx+8 \int \frac {1}{\left (-3+4 x^2\right )^3} \, dx+12 \int \frac {1}{\left (-3+4 x^2\right )^2} \, dx+24 \int \frac {1}{-3+4 x^2} \, dx\\ &=\frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {2 x}{3-4 x^2}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )-2 \int \frac {1}{\left (-3+4 x^2\right )^2} \, dx-2 \int \frac {1}{-3+4 x^2} \, dx\\ &=\frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {5 x}{3 \left (3-4 x^2\right )}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)+\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}}-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )+\frac {1}{3} \int \frac {1}{-3+4 x^2} \, dx\\ &=\frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {5 x}{3 \left (3-4 x^2\right )}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)+\frac {5 \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{6 \sqrt {3}}-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 137, normalized size = 0.85 \[ \frac {\frac {36 x \left (80 x^4-104 x^2+27\right )}{\left (-16 x^6+32 x^4-19 x^2+3\right )^2}-\frac {6 x \left (2288 x^4-2384 x^2+345\right )}{16 x^6-32 x^4+19 x^2-3}-268 \log (1-2 x)+2412 \sqrt {3} \log \left (\sqrt {3}-2 x\right )-3913 \log (1-x)+3913 \log (x+1)+268 \log (2 x+1)-2412 \sqrt {3} \log \left (2 x+\sqrt {3}\right )}{1296} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^(-3),x]

[Out]

((36*x*(27 - 104*x^2 + 80*x^4))/(3 - 19*x^2 + 32*x^4 - 16*x^6)^2 - (6*x*(345 - 2384*x^2 + 2288*x^4))/(-3 + 19*
x^2 - 32*x^4 + 16*x^6) - 268*Log[1 - 2*x] + 2412*Sqrt[3]*Log[Sqrt[3] - 2*x] - 3913*Log[1 - x] + 3913*Log[1 + x
] + 268*Log[1 + 2*x] - 2412*Sqrt[3]*Log[Sqrt[3] + 2*x])/1296

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fricas [B]  time = 0.88, size = 282, normalized size = 1.75 \[ -\frac {219648 \, x^{11} - 668160 \, x^{9} + 751680 \, x^{7} - 382080 \, x^{5} + 85986 \, x^{3} - 2412 \, \sqrt {3} {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (\frac {4 \, x^{2} - 4 \, \sqrt {3} x + 3}{4 \, x^{2} - 3}\right ) - 268 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (2 \, x + 1\right ) + 268 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (2 \, x - 1\right ) - 3913 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (x + 1\right ) + 3913 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (x - 1\right ) - 7182 \, x}{1296 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3)^3,x, algorithm="fricas")

[Out]

-1/1296*(219648*x^11 - 668160*x^9 + 751680*x^7 - 382080*x^5 + 85986*x^3 - 2412*sqrt(3)*(256*x^12 - 1024*x^10 +
1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)*log((4*x^2 - 4*sqrt(3)*x + 3)/(4*x^2 - 3)) - 268*(256*x^12 - 102
4*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)*log(2*x + 1) + 268*(256*x^12 - 1024*x^10 + 1632*x^8 - 13
12*x^6 + 553*x^4 - 114*x^2 + 9)*log(2*x - 1) - 3913*(256*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 11
4*x^2 + 9)*log(x + 1) + 3913*(256*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)*log(x - 1) -
7182*x)/(256*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9)

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giac [A]  time = 0.31, size = 112, normalized size = 0.70 \[ \frac {67}{36} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} \right |}}\right ) - \frac {36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x}{216 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}^{2}} + \frac {67}{324} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {67}{324} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {3913}{1296} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {3913}{1296} \, \log \left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3)^3,x, algorithm="giac")

[Out]

67/36*sqrt(3)*log(abs(8*x - 4*sqrt(3))/abs(8*x + 4*sqrt(3))) - 1/216*(36608*x^11 - 111360*x^9 + 125280*x^7 - 6
3680*x^5 + 14331*x^3 - 1197*x)/(16*x^6 - 32*x^4 + 19*x^2 - 3)^2 + 67/324*log(abs(2*x + 1)) - 67/324*log(abs(2*
x - 1)) + 3913/1296*log(abs(x + 1)) - 3913/1296*log(abs(x - 1))

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maple [A]  time = 0.02, size = 126, normalized size = 0.78 \[ -\frac {67 \sqrt {3}\, \arctanh \left (\frac {2 \sqrt {3}\, x}{3}\right )}{18}-\frac {3913 \ln \left (x -1\right )}{1296}-\frac {67 \ln \left (2 x -1\right )}{324}+\frac {3913 \ln \left (x +1\right )}{1296}+\frac {67 \ln \left (2 x +1\right )}{324}+\frac {1}{432 \left (x -1\right )^{2}}-\frac {67}{432 \left (x -1\right )}+\frac {1}{108 \left (2 x -1\right )^{2}}+\frac {7}{108 \left (2 x -1\right )}-\frac {1}{108 \left (2 x +1\right )^{2}}+\frac {7}{108 \left (2 x +1\right )}-\frac {1}{432 \left (x +1\right )^{2}}-\frac {67}{432 \left (x +1\right )}+\frac {-\frac {20}{3} x^{3}+\frac {13}{3} x}{\left (4 x^{2}-3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-16*x^6+32*x^4-19*x^2+3)^3,x)

[Out]

1/432/(x-1)^2-67/432/(x-1)-3913/1296*ln(x-1)+1/108/(2*x-1)^2+7/108/(2*x-1)-67/324*ln(2*x-1)-1/108/(2*x+1)^2+7/
108/(2*x+1)+67/324*ln(2*x+1)-1/432/(x+1)^2-67/432/(x+1)+3913/1296*ln(x+1)+64*(-5/48*x^3+13/192*x)/(4*x^2-3)^2-
67/18*arctanh(2/3*3^(1/2)*x)*3^(1/2)

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maxima [A]  time = 1.32, size = 119, normalized size = 0.74 \[ \frac {67}{36} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3}}{2 \, x + \sqrt {3}}\right ) - \frac {36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x}{216 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )}} + \frac {67}{324} \, \log \left (2 \, x + 1\right ) - \frac {67}{324} \, \log \left (2 \, x - 1\right ) + \frac {3913}{1296} \, \log \left (x + 1\right ) - \frac {3913}{1296} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x^6+32*x^4-19*x^2+3)^3,x, algorithm="maxima")

[Out]

67/36*sqrt(3)*log((2*x - sqrt(3))/(2*x + sqrt(3))) - 1/216*(36608*x^11 - 111360*x^9 + 125280*x^7 - 63680*x^5 +
14331*x^3 - 1197*x)/(256*x^12 - 1024*x^10 + 1632*x^8 - 1312*x^6 + 553*x^4 - 114*x^2 + 9) + 67/324*log(2*x + 1
) - 67/324*log(2*x - 1) + 3913/1296*log(x + 1) - 3913/1296*log(x - 1)

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mupad [B]  time = 0.09, size = 93, normalized size = 0.58 \[ \frac {-\frac {143\,x^{11}}{216}+\frac {145\,x^9}{72}-\frac {145\,x^7}{64}+\frac {995\,x^5}{864}-\frac {4777\,x^3}{18432}+\frac {133\,x}{6144}}{x^{12}-4\,x^{10}+\frac {51\,x^8}{8}-\frac {41\,x^6}{8}+\frac {553\,x^4}{256}-\frac {57\,x^2}{128}+\frac {9}{256}}-\frac {\mathrm {atan}\left (x\,2{}\mathrm {i}\right )\,67{}\mathrm {i}}{162}-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,3913{}\mathrm {i}}{648}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x\,2{}\mathrm {i}}{3}\right )\,67{}\mathrm {i}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(19*x^2 - 32*x^4 + 16*x^6 - 3)^3,x)

[Out]

((133*x)/6144 - (4777*x^3)/18432 + (995*x^5)/864 - (145*x^7)/64 + (145*x^9)/72 - (143*x^11)/216)/((553*x^4)/25
6 - (57*x^2)/128 - (41*x^6)/8 + (51*x^8)/8 - 4*x^10 + x^12 + 9/256) - (atan(x*2i)*67i)/162 - (atan(x*1i)*3913i
)/648 + (3^(1/2)*atan((3^(1/2)*x*2i)/3)*67i)/18

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sympy [A]  time = 1.45, size = 134, normalized size = 0.83 \[ - \frac {36608 x^{11} - 111360 x^{9} + 125280 x^{7} - 63680 x^{5} + 14331 x^{3} - 1197 x}{55296 x^{12} - 221184 x^{10} + 352512 x^{8} - 283392 x^{6} + 119448 x^{4} - 24624 x^{2} + 1944} - \frac {3913 \log {\left (x - 1 \right )}}{1296} - \frac {67 \log {\left (x - \frac {1}{2} \right )}}{324} + \frac {67 \log {\left (x + \frac {1}{2} \right )}}{324} + \frac {3913 \log {\left (x + 1 \right )}}{1296} + \frac {67 \sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} \right )}}{36} - \frac {67 \sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-16*x**6+32*x**4-19*x**2+3)**3,x)

[Out]

-(36608*x**11 - 111360*x**9 + 125280*x**7 - 63680*x**5 + 14331*x**3 - 1197*x)/(55296*x**12 - 221184*x**10 + 35
2512*x**8 - 283392*x**6 + 119448*x**4 - 24624*x**2 + 1944) - 3913*log(x - 1)/1296 - 67*log(x - 1/2)/324 + 67*l
og(x + 1/2)/324 + 3913*log(x + 1)/1296 + 67*sqrt(3)*log(x - sqrt(3)/2)/36 - 67*sqrt(3)*log(x + sqrt(3)/2)/36

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