3.182 \(\int \frac{1}{2+x^3+x^6} \, dx\)

Optimal. Leaf size=381 \[ \frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt{7}\right )} x+\left (1-i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1-i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt{7}\right )} x+\left (1+i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1+i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}+\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}} \]

[Out]

(I*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt[21]*((1 - I*Sqrt[7])/2)^(2/3)) - (I*ArcTan[(1
- (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt[21]*((1 + I*Sqrt[7])/2)^(2/3)) - ((I/3)*Log[(1 - I*Sqrt[7])
^(1/3) + 2^(1/3)*x])/(Sqrt[7]*((1 - I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 + I*Sqrt[7])^(1/3) + 2^(1/3)*x])/(Sqr
t[7]*((1 + I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 - I*Sqrt[7])^(2/3) - (2*(1 - I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2
])/(2^(1/3)*Sqrt[7]*(1 - I*Sqrt[7])^(2/3)) - ((I/3)*Log[(1 + I*Sqrt[7])^(2/3) - (2*(1 + I*Sqrt[7]))^(1/3)*x +
2^(2/3)*x^2])/(2^(1/3)*Sqrt[7]*(1 + I*Sqrt[7])^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 0.402723, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {1347, 200, 31, 634, 617, 204, 628} \[ \frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt{7}\right )} x+\left (1-i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1-i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt{7}\right )} x+\left (1+i \sqrt{7}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{7} \left (1+i \sqrt{7}\right )^{2/3}}-\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}+\frac{i \log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{7}}\right )}{3 \sqrt{7} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1-i \sqrt{7}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{7}\right )}}}{\sqrt{3}}\right )}{\sqrt{21} \left (\frac{1}{2} \left (1+i \sqrt{7}\right )\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + x^3 + x^6)^(-1),x]

[Out]

(I*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt[21]*((1 - I*Sqrt[7])/2)^(2/3)) - (I*ArcTan[(1
- (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]])/(Sqrt[21]*((1 + I*Sqrt[7])/2)^(2/3)) - ((I/3)*Log[(1 - I*Sqrt[7])
^(1/3) + 2^(1/3)*x])/(Sqrt[7]*((1 - I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 + I*Sqrt[7])^(1/3) + 2^(1/3)*x])/(Sqr
t[7]*((1 + I*Sqrt[7])/2)^(2/3)) + ((I/3)*Log[(1 - I*Sqrt[7])^(2/3) - (2*(1 - I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2
])/(2^(1/3)*Sqrt[7]*(1 - I*Sqrt[7])^(2/3)) - ((I/3)*Log[(1 + I*Sqrt[7])^(2/3) - (2*(1 + I*Sqrt[7]))^(1/3)*x +
2^(2/3)*x^2])/(2^(1/3)*Sqrt[7]*(1 + I*Sqrt[7])^(2/3))

Rule 1347

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

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 31

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

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 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 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 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]

Rubi steps

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

Mathematica [C]  time = 0.0091525, size = 38, normalized size = 0.1 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6+\text{$\#$1}^3+2\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5+\text{$\#$1}^2}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x^3 + x^6)^(-1),x]

[Out]

RootSum[2 + #1^3 + #1^6 & , Log[x - #1]/(#1^2 + 2*#1^5) & ]/3

________________________________________________________________________________________

Maple [C]  time = 0.003, size = 33, normalized size = 0.1 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+{{\it \_Z}}^{3}+2 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}+{{\it \_R}}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6+x^3+2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{6} + x^{3} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6+x^3+2),x, algorithm="maxima")

[Out]

integrate(1/(x^6 + x^3 + 2), x)

________________________________________________________________________________________

Fricas [B]  time = 5.27856, size = 7470, normalized size = 19.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6+x^3+2),x, algorithm="fricas")

[Out]

1/294*112^(1/6)*49^(2/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))*log(112^(1/6)*49^(2/3)*sqrt(7)*x*sin(2/3*arctan(1/
3*sqrt(7) + 4/3)) + 7*112^(1/6)*49^(2/3)*x*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) + 14*49^(1/3)*14^(1/3)*cos(2/3*a
rctan(1/3*sqrt(7) + 4/3))^2 + 14*49^(1/3)*14^(1/3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 98*x^2) - 2/147*112^
(1/6)*49^(2/3)*arctan(1/2744*(14*112^(5/6)*49^(1/3)*sqrt(7)*x*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) + 2744*sqrt(7
)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 2744*sqrt(7)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 98*(112^(5/6)*49^
(1/3)*x + 224*cos(2/3*arctan(1/3*sqrt(7) + 4/3)))*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) - sqrt(112^(1/6)*49^(2/3)
*sqrt(7)*x*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) + 7*112^(1/6)*49^(2/3)*x*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) + 14
*49^(1/3)*14^(1/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 14*49^(1/3)*14^(1/3)*sin(2/3*arctan(1/3*sqrt(7) + 4/
3))^2 + 98*x^2)*(112^(5/6)*49^(1/3)*sqrt(7)*sqrt(2)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) + 7*112^(5/6)*49^(1/3)*
sqrt(2)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))))/(cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 - 7*sin(2/3*arctan(1/3*sqrt
(7) + 4/3))^2))*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) + 1/147*(112^(1/6)*49^(2/3)*sqrt(3)*cos(2/3*arctan(1/3*sqrt
(7) + 4/3)) + 112^(1/6)*49^(2/3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3)))*arctan(1/5488*(70*112^(5/6)*49^(1/3)*(sqr
t(7)*x + 7*sqrt(3)*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^3 - 27440*(sqrt(7) + 2*sqrt(3))*cos(2/3*arctan(1/3*sq
rt(7) + 4/3))^4 - 5488*(sqrt(7) - 2*sqrt(3))*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^4 - 14*(112^(5/6)*49^(1/3)*(sq
rt(7)*sqrt(3)*x - 7*x) - 1568*(sqrt(7)*sqrt(3) - 5)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)))*sin(2/3*arctan(1/3*sqr
t(7) + 4/3))^3 + 14*(112^(5/6)*49^(1/3)*(13*sqrt(7)*x - 21*sqrt(3)*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) - 784
*(3*sqrt(7) + 4*sqrt(3))*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 - 14*(112^
(5/6)*49^(1/3)*(9*sqrt(7)*sqrt(3)*x + 49*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 - 1568*(sqrt(7)*sqrt(3) + 11)
*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) - (5*112^(5/6)*49^(1/3)*(sqrt(7) + 7
*sqrt(3))*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^3 - 112^(5/6)*49^(1/3)*(9*sqrt(7)*sqrt(3) + 49)*cos(2/3*arctan(1/
3*sqrt(7) + 4/3))^2*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) + 112^(5/6)*49^(1/3)*(13*sqrt(7) - 21*sqrt(3))*cos(2/3*
arctan(1/3*sqrt(7) + 4/3))*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 - 112^(5/6)*49^(1/3)*(sqrt(7)*sqrt(3) - 7)*sin
(2/3*arctan(1/3*sqrt(7) + 4/3))^3)*sqrt(-112^(1/6)*49^(2/3)*(sqrt(7)*sqrt(3)*x + 7*x)*cos(2/3*arctan(1/3*sqrt(
7) + 4/3)) - 112^(1/6)*49^(2/3)*(sqrt(7)*x - 7*sqrt(3)*x)*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) + 28*49^(1/3)*14^
(1/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 28*49^(1/3)*14^(1/3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 196*x
^2))/(25*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^4 - 38*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2*sin(2/3*arctan(1/3*sqr
t(7) + 4/3))^2 + sin(2/3*arctan(1/3*sqrt(7) + 4/3))^4)) + 1/147*(112^(1/6)*49^(2/3)*sqrt(3)*cos(2/3*arctan(1/3
*sqrt(7) + 4/3)) - 112^(1/6)*49^(2/3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3)))*arctan(-1/5488*(70*112^(5/6)*49^(1/3
)*(sqrt(7)*x - 7*sqrt(3)*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^3 - 27440*(sqrt(7) - 2*sqrt(3))*cos(2/3*arctan(
1/3*sqrt(7) + 4/3))^4 - 5488*(sqrt(7) + 2*sqrt(3))*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^4 + 14*(112^(5/6)*49^(1/
3)*(sqrt(7)*sqrt(3)*x + 7*x) - 1568*(sqrt(7)*sqrt(3) + 5)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)))*sin(2/3*arctan(1
/3*sqrt(7) + 4/3))^3 + 14*(112^(5/6)*49^(1/3)*(13*sqrt(7)*x + 21*sqrt(3)*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))
- 784*(3*sqrt(7) - 4*sqrt(3))*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 14
*(112^(5/6)*49^(1/3)*(9*sqrt(7)*sqrt(3)*x - 49*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 - 1568*(sqrt(7)*sqrt(3)
- 11)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) - (5*112^(5/6)*49^(1/3)*(sqrt(
7) - 7*sqrt(3))*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^3 + 112^(5/6)*49^(1/3)*(9*sqrt(7)*sqrt(3) - 49)*cos(2/3*arc
tan(1/3*sqrt(7) + 4/3))^2*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) + 112^(5/6)*49^(1/3)*(13*sqrt(7) + 21*sqrt(3))*co
s(2/3*arctan(1/3*sqrt(7) + 4/3))*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 112^(5/6)*49^(1/3)*(sqrt(7)*sqrt(3) +
7)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^3)*sqrt(112^(1/6)*49^(2/3)*(sqrt(7)*sqrt(3)*x - 7*x)*cos(2/3*arctan(1/3*
sqrt(7) + 4/3)) - 112^(1/6)*49^(2/3)*(sqrt(7)*x + 7*sqrt(3)*x)*sin(2/3*arctan(1/3*sqrt(7) + 4/3)) + 28*49^(1/3
)*14^(1/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 28*49^(1/3)*14^(1/3)*sin(2/3*arctan(1/3*sqrt(7) + 4/3))^2 +
196*x^2))/(25*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^4 - 38*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2*sin(2/3*arctan(1/
3*sqrt(7) + 4/3))^2 + sin(2/3*arctan(1/3*sqrt(7) + 4/3))^4)) + 1/588*(112^(1/6)*49^(2/3)*sqrt(3)*sin(2/3*arcta
n(1/3*sqrt(7) + 4/3)) - 112^(1/6)*49^(2/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)))*log(-112^(1/6)*49^(2/3)*(sqrt(7
)*sqrt(3)*x + 7*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) - 112^(1/6)*49^(2/3)*(sqrt(7)*x - 7*sqrt(3)*x)*sin(2/3*a
rctan(1/3*sqrt(7) + 4/3)) + 28*49^(1/3)*14^(1/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 28*49^(1/3)*14^(1/3)*s
in(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 196*x^2) - 1/588*(112^(1/6)*49^(2/3)*sqrt(3)*sin(2/3*arctan(1/3*sqrt(7)
+ 4/3)) + 112^(1/6)*49^(2/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)))*log(112^(1/6)*49^(2/3)*(sqrt(7)*sqrt(3)*x - 7
*x)*cos(2/3*arctan(1/3*sqrt(7) + 4/3)) - 112^(1/6)*49^(2/3)*(sqrt(7)*x + 7*sqrt(3)*x)*sin(2/3*arctan(1/3*sqrt(
7) + 4/3)) + 28*49^(1/3)*14^(1/3)*cos(2/3*arctan(1/3*sqrt(7) + 4/3))^2 + 28*49^(1/3)*14^(1/3)*sin(2/3*arctan(1
/3*sqrt(7) + 4/3))^2 + 196*x^2)

________________________________________________________________________________________

Sympy [A]  time = 0.153661, size = 24, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (1000188 t^{6} + 1323 t^{3} + 1, \left ( t \mapsto t \log{\left (- 5292 t^{4} + 7 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**6+x**3+2),x)

[Out]

RootSum(1000188*_t**6 + 1323*_t**3 + 1, Lambda(_t, _t*log(-5292*_t**4 + 7*_t + x)))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{6} + x^{3} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^6+x^3+2),x, algorithm="giac")

[Out]

integrate(1/(x^6 + x^3 + 2), x)