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]

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

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Rubi [A]  time = 0.40, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, 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 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 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]

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*}

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Mathematica [C]  time = 0.01, size = 38, normalized size = 0.10 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^3+2\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^5+\text {$\#$1}^2}\& \right ] \]

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

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fricas [B]  time = 2.57, size = 1996, normalized size = 5.24 \[ \text {result too large to display} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{6} + x^{3} + 2}\,{d x} \]

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)

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maple [C]  time = 0.01, size = 33, normalized size = 0.09 \[ \frac {\ln \left (-\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )^{5}+3 \RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sum(1/(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 \[ \int \frac {1}{x^{6} + x^{3} + 2}\,{d x} \]

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)

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mupad [B]  time = 2.61, size = 513, normalized size = 1.35 \[ \frac {\ln \left (x+\frac {7^{1/3}\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{4}+\frac {7^{5/6}\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}\right )\,{\left (-49-\sqrt {7}\,21{}\mathrm {i}\right )}^{1/3}}{42}+\frac {\ln \left (x+\frac {7^{1/3}\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{4}-\frac {7^{5/6}\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}\right )\,{\left (-49+\sqrt {7}\,21{}\mathrm {i}\right )}^{1/3}}{42}+\frac {7^{1/3}\,\ln \left (6\,x+\frac {7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x+\frac {567\,7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84}+\frac {7^{1/3}\,\ln \left (6\,x+\frac {7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x+\frac {567\,7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84}-\frac {7^{1/3}\,\ln \left (6\,x-\frac {7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x-\frac {567\,7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84}-\frac {7^{1/3}\,\ln \left (6\,x-\frac {7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x-\frac {567\,7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3 + x^6 + 2),x)

[Out]

(log(x + (7^(1/3)*(- 7^(1/2)*3i - 7)^(1/3))/4 + (7^(5/6)*(- 7^(1/2)*3i - 7)^(1/3)*1i)/28)*(- 7^(1/2)*21i - 49)
^(1/3))/42 + (log(x + (7^(1/3)*(7^(1/2)*3i - 7)^(1/3))/4 - (7^(5/6)*(7^(1/2)*3i - 7)^(1/3)*1i)/28)*(7^(1/2)*21
i - 49)^(1/3))/42 + (7^(1/3)*log(6*x + (7^(1/3)*(3^(1/2)*1i - 1)*(- 7^(1/2)*3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1
i - 1)^2*(- 7^(1/2)*3i - 7)^(2/3)*(3969*x + (567*7^(1/3)*(3^(1/2)*1i - 1)*(- 7^(1/2)*3i - 7)^(1/3))/2))/7056 +
63))/84)*(3^(1/2)*1i - 1)*(- 7^(1/2)*3i - 7)^(1/3))/84 + (7^(1/3)*log(6*x + (7^(1/3)*(3^(1/2)*1i - 1)*(7^(1/2
)*3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1i - 1)^2*(7^(1/2)*3i - 7)^(2/3)*(3969*x + (567*7^(1/3)*(3^(1/2)*1i - 1)*(7
^(1/2)*3i - 7)^(1/3))/2))/7056 + 63))/84)*(3^(1/2)*1i - 1)*(7^(1/2)*3i - 7)^(1/3))/84 - (7^(1/3)*log(6*x - (7^
(1/3)*(3^(1/2)*1i + 1)*(- 7^(1/2)*3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1i + 1)^2*(- 7^(1/2)*3i - 7)^(2/3)*(3969*x
- (567*7^(1/3)*(3^(1/2)*1i + 1)*(- 7^(1/2)*3i - 7)^(1/3))/2))/7056 + 63))/84)*(3^(1/2)*1i + 1)*(- 7^(1/2)*3i -
7)^(1/3))/84 - (7^(1/3)*log(6*x - (7^(1/3)*(3^(1/2)*1i + 1)*(7^(1/2)*3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1i + 1)
^2*(7^(1/2)*3i - 7)^(2/3)*(3969*x - (567*7^(1/3)*(3^(1/2)*1i + 1)*(7^(1/2)*3i - 7)^(1/3))/2))/7056 + 63))/84)*
(3^(1/2)*1i + 1)*(7^(1/2)*3i - 7)^(1/3))/84

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sympy [A]  time = 0.15, size = 24, normalized size = 0.06 \[ \operatorname {RootSum} {\left (1000188 t^{6} + 1323 t^{3} + 1, \left (t \mapsto t \log {\left (- 5292 t^{4} + 7 t + x \right )} \right )\right )} \]

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)))

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