3.184 \(\int \frac {x^3}{2+x^3+x^6} \, dx\)

Optimal. Leaf size=399 \[ -\frac {\left (7+i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+\left (1-i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+\left (1+i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}} \]

[Out]

1/42*ln(2^(1/3)*x+(1+I*7^(1/2))^(1/3))*(7-I*7^(1/2))*2^(2/3)/(1+I*7^(1/2))^(2/3)-1/84*ln(2^(2/3)*x^2-2^(1/3)*x
*(1+I*7^(1/2))^(1/3)+(1+I*7^(1/2))^(2/3))*(7-I*7^(1/2))*2^(2/3)/(1+I*7^(1/2))^(2/3)+1/42*ln(2^(1/3)*x+(1-I*7^(
1/2))^(1/3))*(7+I*7^(1/2))*2^(2/3)/(1-I*7^(1/2))^(2/3)-1/84*ln(2^(2/3)*x^2-2^(1/3)*x*(1-I*7^(1/2))^(1/3)+(1-I*
7^(1/2))^(2/3))*(7+I*7^(1/2))*2^(2/3)/(1-I*7^(1/2))^(2/3)-1/42*I*arctan(1/3*(1-2*2^(1/3)*x/(1-I*7^(1/2))^(1/3)
)*3^(1/2))*(1-I*7^(1/2))^(1/3)*2^(2/3)*21^(1/2)+1/42*I*arctan(1/3*(1-2*2^(1/3)*x/(1+I*7^(1/2))^(1/3))*3^(1/2))
*(1+I*7^(1/2))^(1/3)*2^(2/3)*21^(1/2)

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Rubi [A]  time = 0.31, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1374, 200, 31, 634, 617, 204, 628} \[ -\frac {\left (7+i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+\left (1-i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+\left (1+i \sqrt {7}\right )^{2/3}\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(2 + x^3 + x^6),x]

[Out]

((-I)*((1 - I*Sqrt[7])/2)^(1/3)*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/Sqrt[21] + (I*((1 + I*S
qrt[7])/2)^(1/3)*ArcTan[(1 - (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/Sqrt[21] + ((7 + I*Sqrt[7])*Log[(1 - I
*Sqrt[7])^(1/3) + 2^(1/3)*x])/(21*2^(1/3)*(1 - I*Sqrt[7])^(2/3)) + ((7 - I*Sqrt[7])*Log[(1 + I*Sqrt[7])^(1/3)
+ 2^(1/3)*x])/(21*2^(1/3)*(1 + I*Sqrt[7])^(2/3)) - ((7 + I*Sqrt[7])*Log[(1 - I*Sqrt[7])^(2/3) - (2*(1 - I*Sqrt
[7]))^(1/3)*x + 2^(2/3)*x^2])/(42*2^(1/3)*(1 - I*Sqrt[7])^(2/3)) - ((7 - I*Sqrt[7])*Log[(1 + I*Sqrt[7])^(2/3)
- (2*(1 + I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2])/(42*2^(1/3)*(1 + I*Sqrt[7])^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1374

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rubi steps

\begin {align*} \int \frac {x^3}{2+x^3+x^6} \, dx &=\frac {1}{14} \left (7-i \sqrt {7}\right ) \int \frac {1}{\frac {1}{2}+\frac {i \sqrt {7}}{2}+x^3} \, dx+\frac {1}{14} \left (7+i \sqrt {7}\right ) \int \frac {1}{\frac {1}{2}-\frac {i \sqrt {7}}{2}+x^3} \, dx\\ &=\frac {\left (7-i \sqrt {7}\right ) \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}+x} \, dx}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}+x} \, dx}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}\\ &=\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+x^2} \, dx}{14\ 2^{2/3} \sqrt [3]{1+i \sqrt {7}}}-\frac {\left (7+i \sqrt {7}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+x^2} \, dx}{14\ 2^{2/3} \sqrt [3]{1-i \sqrt {7}}}\\ &=\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{7 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}+\frac {\left (7+i \sqrt {7}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{7 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}\\ &=-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 37, normalized size = 0.09 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^3+2\& ,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{2 \text {$\#$1}^3+1}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(2 + x^3 + x^6),x]

[Out]

RootSum[2 + #1^3 + #1^6 & , (Log[x - #1]*#1)/(1 + 2*#1^3) & ]/3

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fricas [B]  time = 0.98, size = 1435, normalized size = 3.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+x^3+2),x, algorithm="fricas")

[Out]

1/294*98^(2/3)*56^(1/6)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))*log(-2*98^(2/3)*56^(1/6)*sqrt(7)*x*sin
(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 14*98^(1/3)*7^(1/3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(
7)))^2 + 14*98^(1/3)*7^(1/3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 98*x^2) - 2/147*98^(2/3)*56^(
1/6)*arctan(1/5488*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(2)*sqrt(-2*98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*s
qrt(14)*sqrt(7) + sqrt(7))) + 14*98^(1/3)*7^(1/3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 14*98^(1
/3)*7^(1/3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 98*x^2) - 14*98^(1/3)*56^(5/6)*sqrt(7)*x + 548
8*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))/cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*sin(2/3*ar
ctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 1/147*(98^(2/3)*56^(1/6)*sqrt(3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) +
sqrt(7))) + 98^(2/3)*56^(1/6)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*arctan(1/2744*(14*98^(1/3)*56^
(5/6)*sqrt(7)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 2744*sqrt(3)*cos(2/3*arctan(2/7*sqrt(14)*sqr
t(7) + sqrt(7)))^2 + 2744*sqrt(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 14*(98^(1/3)*56^(5/6)*sq
rt(7)*sqrt(3)*x + 784*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + s
qrt(7))) - sqrt(98^(2/3)*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(2/3)
*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 14*98^(1/3)*7^(1/3)*cos(2/3*arctan(2/7*s
qrt(14)*sqrt(7) + sqrt(7)))^2 + 14*98^(1/3)*7^(1/3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 98*x^2
)*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(3)*sqrt(2)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(1/3)*56^(5/
6)*sqrt(7)*sqrt(2)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))))/(cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sq
rt(7)))^2 - 3*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2)) + 1/147*(98^(2/3)*56^(1/6)*sqrt(3)*cos(2/3*a
rctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) - 98^(2/3)*56^(1/6)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*ar
ctan(-1/2744*(14*98^(1/3)*56^(5/6)*sqrt(7)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) - 2744*sqrt(3)*co
s(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 - 2744*sqrt(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))
^2 - 14*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(3)*x - 784*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*sin(2/3*ar
ctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + sqrt(-98^(2/3)*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*
sqrt(7) + sqrt(7))) + 98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 14*98^(1/3
)*7^(1/3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 14*98^(1/3)*7^(1/3)*sin(2/3*arctan(2/7*sqrt(14)*
sqrt(7) + sqrt(7)))^2 + 98*x^2)*(98^(1/3)*56^(5/6)*sqrt(7)*sqrt(3)*sqrt(2)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7)
+ sqrt(7))) - 98^(1/3)*56^(5/6)*sqrt(7)*sqrt(2)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))))/(cos(2/3*ar
ctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 - 3*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2)) + 1/588*(98^(2
/3)*56^(1/6)*sqrt(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) - 98^(2/3)*56^(1/6)*cos(2/3*arctan(2/7*sq
rt(14)*sqrt(7) + sqrt(7))))*log(98^(2/3)*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt
(7))) + 98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 14*98^(1/3)*7^(1/3)*cos(
2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 14*98^(1/3)*7^(1/3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt
(7)))^2 + 98*x^2) - 1/588*(98^(2/3)*56^(1/6)*sqrt(3)*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(2/3
)*56^(1/6)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))))*log(-98^(2/3)*56^(1/6)*sqrt(7)*sqrt(3)*x*cos(2/3*a
rctan(2/7*sqrt(14)*sqrt(7) + sqrt(7))) + 98^(2/3)*56^(1/6)*sqrt(7)*x*sin(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqr
t(7))) + 14*98^(1/3)*7^(1/3)*cos(2/3*arctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 14*98^(1/3)*7^(1/3)*sin(2/3*a
rctan(2/7*sqrt(14)*sqrt(7) + sqrt(7)))^2 + 98*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{x^{6} + x^{3} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+x^3+2),x, algorithm="giac")

[Out]

integrate(x^3/(x^6 + x^3 + 2), x)

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maple [C]  time = 0.01, size = 36, normalized size = 0.09 \[ \frac {\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )^{3} \ln \left (-\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )^{5}+3 \RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^6+x^3+2),x)

[Out]

1/3*sum(_R^3/(2*_R^5+_R^2)*ln(-_R+x),_R=RootOf(_Z^6+_Z^3+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{x^{6} + x^{3} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^6+x^3+2),x, algorithm="maxima")

[Out]

integrate(x^3/(x^6 + x^3 + 2), x)

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mupad [B]  time = 2.61, size = 351, normalized size = 0.88 \[ \frac {\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-196-\sqrt {7}\,28{}\mathrm {i}\right )}^{1/3}}{42}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{42}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^3 + x^6 + 2),x)

[Out]

(log(x - (2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i)/14)*(- 7^(1/2)*28i - 196)^(1/3))/42 + (2^(2/3)*7^(1/3)*
log(x + (2^(2/3)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3)*1i)/14)*(7^(1/2)*1i - 7)^(1/3))/42 - (2^(2/3)*7^(1/3)*log(x +
(2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i)/28 - (2^(2/3)*3^(1/2)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3))/28)*(3^(
1/2)*1i + 1)*(- 7^(1/2)*1i - 7)^(1/3))/84 + (2^(2/3)*7^(1/3)*log(x + (2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)
*1i)/28 + (2^(2/3)*3^(1/2)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i - 1)*(- 7^(1/2)*1i - 7)^(1/3))/84
+ (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3)*1i)/28 - (2^(2/3)*3^(1/2)*7^(5/6)*(7^(1/2)
*1i - 7)^(1/3))/28)*(3^(1/2)*1i - 1)*(7^(1/2)*1i - 7)^(1/3))/84 - (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(5/6)*(7
^(1/2)*1i - 7)^(1/3)*1i)/28 + (2^(2/3)*3^(1/2)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i + 1)*(7^(1/2)*1
i - 7)^(1/3))/84

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sympy [A]  time = 0.15, size = 24, normalized size = 0.06 \[ \operatorname {RootSum} {\left (250047 t^{6} + 1323 t^{3} + 2, \left (t \mapsto t \log {\left (7938 t^{4} + 21 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**6+x**3+2),x)

[Out]

RootSum(250047*_t**6 + 1323*_t**3 + 2, Lambda(_t, _t*log(7938*_t**4 + 21*_t + x)))

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