3.1.67 \(\int \frac {2+x-x^3}{\sqrt {1+x+x^3} (1+x-x^2+x^3)} \, dx\)

Optimal. Leaf size=15 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^3+x+1}}\right ) \]

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Rubi [F]  time = 2.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]

[Out]

(((-2*I)/3)*Sqrt[(2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3) + 6^(2/3)*x)/(6*(3/(-9 + Sqrt[93]))^
(1/3) - 3*(2*(-9 + Sqrt[93]))^(1/3) - I*6^(1/6)*Sqrt[12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9
+ Sqrt[93]))^(2/3)])]*Sqrt[6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3) - 6*3^
(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2]*EllipticF[ArcSin[Sqrt[I*(6^(1/3)*(2*
(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3) - I*6^(1/6)*Sqrt[12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3
) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3)]) - 12*x)]/(2^(3/4)*(3*(12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/
3)*(3*(-9 + Sqrt[93]))^(2/3)))^(1/4))], (2*6^(1/6)*Sqrt[12 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*
(-9 + Sqrt[93]))^(2/3)])/(I*(6*(3/(-9 + Sqrt[93]))^(1/3) - 3*(2*(-9 + Sqrt[93]))^(1/3)) + 6^(1/6)*Sqrt[12 + 6*
3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3)])])/Sqrt[1 + x + x^3] + 3*Defer[Int][1/(
Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)), x] + 2*Defer[Int][x/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)), x] - Defe
r[Int][x^2/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)), x]

Rubi steps

\begin {align*} \int \frac {2+x-x^3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx &=\int \left (-\frac {1}{\sqrt {1+x+x^3}}+\frac {3+2 x-x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}\right ) \, dx\\ &=-\int \frac {1}{\sqrt {1+x+x^3}} \, dx+\int \frac {3+2 x-x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx\\ &=-\frac {\left (\sqrt {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt {\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}\right ) \int \frac {1}{\sqrt {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt {\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}} \, dx}{\sqrt {1+x+x^3}}+\int \left (\frac {3}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}+\frac {2 x}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}-\frac {x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {x}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx+3 \int \frac {1}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx-\frac {\left (2 \sqrt {\frac {1}{3} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )} \sqrt {\frac {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x}{\frac {\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}}{3^{2/3}}+\frac {\sqrt [3]{2} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )}{3^{2/3}}-\sqrt {\frac {1}{6} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}} \sqrt {-\frac {\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}{\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right )^2}{3 \sqrt [3]{3}}-\frac {2}{9} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {\sqrt {\frac {2}{3} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )} x^2}{\frac {\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}}{3^{2/3}}+\frac {\sqrt [3]{2} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )}{3^{2/3}}-\sqrt {\frac {1}{6} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}}} \, dx,x,\sqrt {\frac {-\frac {\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}}{3^{2/3}}+\sqrt {\frac {1}{6} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}+2 x}{\sqrt {\frac {2}{3} \left (-12-6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}}\right )}{\sqrt {1+x+x^3}}-\int \frac {x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx\\ &=-\frac {2 i \sqrt {\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}+6^{2/3} x}{6 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-3 \sqrt [3]{2 \left (-9+\sqrt {93}\right )}-i \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}} \sqrt {6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}-6 \sqrt [3]{3} \left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x+18 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {i \left (\sqrt [3]{6} \left (2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}-i \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}\right )-12 x\right )}}{2^{3/4} \sqrt [4]{3 \left (12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )}}\right )|\frac {2 \sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}{i \left (6 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-3 \sqrt [3]{2 \left (-9+\sqrt {93}\right )}\right )+\sqrt [6]{6} \sqrt {12+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}}}\right )}{3 \sqrt {1+x+x^3}}+2 \int \frac {x}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx+3 \int \frac {1}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx-\int \frac {x^2}{\sqrt {1+x+x^3} \left (1+x-x^2+x^3\right )} \, dx\\ \end {align*}

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Mathematica [C]  time = 3.52, size = 3866, normalized size = 257.73 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]

[Out]

(2*Sqrt[(-x + Root[1 + #1 + #1^3 & , 1, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])]*(-(
(EllipticF[ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3
& , 3, 0])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - R
oot[1 + #1 + #1^3 & , 3, 0])]*(x - Root[1 + #1 + #1^3 & , 3, 0])*Sqrt[(-x + Root[1 + #1 + #1^3 & , 2, 0])/(Roo
t[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])])/Sqrt[(x - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 +
#1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])]) + (2*EllipticPi[(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1
+ #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 1, 0]), ArcSin[Sqrt[(-x +
Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])]], (Root[1 + #1
+ #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])
]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #1^3 & , 3, 0]))/(Root[1 + #1 + #1^3 & , 2, 0]
- Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0]))/((Root[1 +
#1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 1, 0])*(Root[1 + #1 - #1^2 + #1^3 & , 1, 0] - Root[1 + #1
- #1^2 + #1^3 & , 2, 0])*(Root[1 + #1 - #1^2 + #1^3 & , 1, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) + (Elli
pticPi[(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 +
#1 - #1^2 + #1^3 & , 1, 0]), ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] +
Root[1 + #1 + #1^3 & , 3, 0])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 +
#1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 +
#1^3 & , 3, 0]))/(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2,
0] - Root[1 + #1 + #1^3 & , 3, 0])*Root[1 + #1 - #1^2 + #1^3 & , 1, 0])/((Root[1 + #1 + #1^3 & , 3, 0] - Root
[1 + #1 - #1^2 + #1^3 & , 1, 0])*(Root[1 + #1 - #1^2 + #1^3 & , 1, 0] - Root[1 + #1 - #1^2 + #1^3 & , 2, 0])*(
Root[1 + #1 - #1^2 + #1^3 & , 1, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) - (EllipticPi[(-Root[1 + #1 + #1^3
& , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 1, 0]
), ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3,
0])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 +
#1 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #1^3 & , 3, 0]))/(Root[1 +
#1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 &
, 3, 0])*Root[1 + #1 - #1^2 + #1^3 & , 1, 0]^3)/((Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & ,
1, 0])*(Root[1 + #1 - #1^2 + #1^3 & , 1, 0] - Root[1 + #1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + #1 - #1^2 + #1^3
& , 1, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) + (2*EllipticPi[(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #
1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 2, 0]), ArcSin[Sqrt[(-x + Ro
ot[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])]], (Root[1 + #1 + #
1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])]*S
qrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #1^3 & , 3, 0]))/(Root[1 + #1 + #1^3 & , 2, 0] -
Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0]))/((Root[1 + #1
+ #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 2, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 1, 0] + Root[1 + #1 -
#1^2 + #1^3 & , 2, 0])*(Root[1 + #1 - #1^2 + #1^3 & , 2, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) + (Ellipt
icPi[(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #
1 - #1^2 + #1^3 & , 2, 0]), ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + R
oot[1 + #1 + #1^3 & , 3, 0])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #
1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #
1^3 & , 3, 0]))/(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0
] - Root[1 + #1 + #1^3 & , 3, 0])*Root[1 + #1 - #1^2 + #1^3 & , 2, 0])/((Root[1 + #1 + #1^3 & , 3, 0] - Root[1
+ #1 - #1^2 + #1^3 & , 2, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 1, 0] + Root[1 + #1 - #1^2 + #1^3 & , 2, 0])*(R
oot[1 + #1 - #1^2 + #1^3 & , 2, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) - (EllipticPi[(-Root[1 + #1 + #1^3
& , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 2, 0])
, ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0
])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 + #
1 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #1^3 & , 3, 0]))/(Root[1 +
#1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & ,
3, 0])*Root[1 + #1 - #1^2 + #1^3 & , 2, 0]^3)/((Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & ,
2, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 1, 0] + Root[1 + #1 - #1^2 + #1^3 & , 2, 0])*(Root[1 + #1 - #1^2 + #1^3
& , 2, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) + (2*EllipticPi[(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #
1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[(-x + Ro
ot[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])]], (Root[1 + #1 + #
1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])]*S
qrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #1^3 & , 3, 0]))/(Root[1 + #1 + #1^3 & , 2, 0] -
Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0]))/((Root[1 + #1
+ #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 1, 0] + Root[1 + #1 -
#1^2 + #1^3 & , 3, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 2, 0] + Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) + (Ellip
ticPi[(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 +
#1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] +
Root[1 + #1 + #1^3 & , 3, 0])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 +
#1^3 & , 1, 0] - Root[1 + #1 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 +
#1^3 & , 3, 0]))/(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2,
0] - Root[1 + #1 + #1^3 & , 3, 0])*Root[1 + #1 - #1^2 + #1^3 & , 3, 0])/((Root[1 + #1 + #1^3 & , 3, 0] - Root[
1 + #1 - #1^2 + #1^3 & , 3, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 1, 0] + Root[1 + #1 - #1^2 + #1^3 & , 3, 0])*(
-Root[1 + #1 - #1^2 + #1^3 & , 2, 0] + Root[1 + #1 - #1^2 + #1^3 & , 3, 0])) - (EllipticPi[(-Root[1 + #1 + #1^
3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 & , 3, 0
]), ArcSin[Sqrt[(-x + Root[1 + #1 + #1^3 & , 3, 0])/(-Root[1 + #1 + #1^3 & , 2, 0] + Root[1 + #1 + #1^3 & , 3,
0])]], (Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])/(Root[1 + #1 + #1^3 & , 1, 0] - Root[1 +
#1 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 + #1 + #1^3 & , 2, 0])*(x - Root[1 + #1 + #1^3 & , 3, 0]))/(Root[1
+ #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 & , 3, 0])^2)]*(Root[1 + #1 + #1^3 & , 2, 0] - Root[1 + #1 + #1^3 &
, 3, 0])*Root[1 + #1 - #1^2 + #1^3 & , 3, 0]^3)/((Root[1 + #1 + #1^3 & , 3, 0] - Root[1 + #1 - #1^2 + #1^3 &
, 3, 0])*(-Root[1 + #1 - #1^2 + #1^3 & , 1, 0] + Root[1 + #1 - #1^2 + #1^3 & , 3, 0])*(-Root[1 + #1 - #1^2 + #
1^3 & , 2, 0] + Root[1 + #1 - #1^2 + #1^3 & , 3, 0]))))/Sqrt[1 + x + x^3]

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IntegrateAlgebraic [A]  time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^3+x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x - x^3)/(Sqrt[1 + x + x^3]*(1 + x - x^2 + x^3)),x]

[Out]

2*ArcTanh[x/Sqrt[1 + x + x^3]]

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fricas [B]  time = 0.42, size = 35, normalized size = 2.33 \begin {gather*} \log \left (\frac {x^{3} + x^{2} + 2 \, \sqrt {x^{3} + x + 1} x + x + 1}{x^{3} - x^{2} + x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x^3 + x^2 + 2*sqrt(x^3 + x + 1)*x + x + 1)/(x^3 - x^2 + x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{3} - x - 2}{{\left (x^{3} - x^{2} + x + 1\right )} \sqrt {x^{3} + x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(x^3 - x - 2)/((x^3 - x^2 + x + 1)*sqrt(x^3 + x + 1)), x)

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maple [C]  time = 2.61, size = 1905, normalized size = 127.00 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2))^(1/3))*(I*(x-1/12*(108+12*93^(1/2))^(1/3)+1/(
108+12*93^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2))^(1/3)))*3^(1/2)/(1/6*(1
08+12*93^(1/2))^(1/3)+2/(108+12*93^(1/2))^(1/3)))^(1/2)*((x+1/6*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2))^(1
/3))/(1/4*(108+12*93^(1/2))^(1/3)-3/(108+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108
+12*93^(1/2))^(1/3))))^(1/2)*(-I*(x-1/12*(108+12*93^(1/2))^(1/3)+1/(108+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/6
*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2))^(1/3)))*3^(1/2)/(1/6*(108+12*93^(1/2))^(1/3)+2/(108+12*93^(1/2))^
(1/3)))^(1/2)/(x^3+x+1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/12*(108+12*93^(1/2))^(1/3)+1/(108+12*93^(1/2))^(1/
3)+1/2*I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2))^(1/3)))*3^(1/2)/(1/6*(108+12*93^(1/2))^(1/3
)+2/(108+12*93^(1/2))^(1/3)))^(1/2),(I*3^(1/2)*(1/6*(108+12*93^(1/2))^(1/3)+2/(108+12*93^(1/2))^(1/3))/(1/4*(1
08+12*93^(1/2))^(1/3)-3/(108+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2)
)^(1/3))))^(1/2))-1/216*I*6^(1/2)*12^(1/3)*sum(_alpha*(-(9+93^(1/2))^(1/3)-12^(1/3)/(9+93^(1/2))^(1/3))*(1/24*
I*(12*x+12^(1/3)*(-(9+93^(1/2))^(1/3)+12^(1/3)/(9+93^(1/2))^(1/3)+I*3^(1/2)*(-(9+93^(1/2))^(1/3)-12^(1/3)/(9+9
3^(1/2))^(1/3))))/((9+93^(1/2))^(1/3)+12^(1/3)/(9+93^(1/2))^(1/3)))^(1/2)*((6*x+12^(1/3)*((9+93^(1/2))^(1/3)-1
2^(1/3)/(9+93^(1/2))^(1/3)))/(3*(9+93^(1/2))^(1/3)-3*12^(1/3)/(9+93^(1/2))^(1/3)-I*3^(1/2)*(-(9+93^(1/2))^(1/3
)-12^(1/3)/(9+93^(1/2))^(1/3))))^(1/2)*(-1/24*I*(12*x+12^(1/3)*(-(9+93^(1/2))^(1/3)+12^(1/3)/(9+93^(1/2))^(1/3
)-I*3^(1/2)*(-(9+93^(1/2))^(1/3)-12^(1/3)/(9+93^(1/2))^(1/3))))/((9+93^(1/2))^(1/3)+12^(1/3)/(9+93^(1/2))^(1/3
)))^(1/2)/(x^3+x+1)^(1/2)*(-48*_alpha^2+96*_alpha-240+12^(1/3)*(31^(1/2)*3^(1/2)*12^(1/3)*_alpha*(3^(1/2)*31^(
1/2)+9)^(2/3)+2*3^(1/2)*31^(1/2)*_alpha^2*(3^(1/2)*31^(1/2)+9)^(1/3)-31^(1/2)*3^(1/2)*12^(1/3)*_alpha^2*(3^(1/
2)*31^(1/2)+9)^(2/3)-4*3^(1/2)*31^(1/2)*_alpha*(3^(1/2)*31^(1/2)+9)^(1/3)+3*I*12^(1/3)*31^(1/2)*(3^(1/2)*31^(1
/2)+9)^(2/3)-13*I*3^(1/2)*12^(1/3)*(3^(1/2)*31^(1/2)+9)^(2/3)-3*I*12^(1/3)*31^(1/2)*_alpha*(3^(1/2)*31^(1/2)+9
)^(2/3)+24*I*3^(1/2)*_alpha*(3^(1/2)*31^(1/2)+9)^(1/3)-11*I*3^(1/2)*12^(1/3)*_alpha^2*(3^(1/2)*31^(1/2)+9)^(2/
3)-12*I*31^(1/2)*_alpha*(3^(1/2)*31^(1/2)+9)^(1/3)-6*I*3^(1/2)*_alpha^2*(3^(1/2)*31^(1/2)+9)^(1/3)-24*I*3^(1/2
)*(3^(1/2)*31^(1/2)+9)^(1/3)-31^(1/2)*3^(1/2)*12^(1/3)*(3^(1/2)*31^(1/2)+9)^(2/3)+13*12^(1/3)*(3^(1/2)*31^(1/2
)+9)^(2/3)+6*I*31^(1/2)*_alpha^2*(3^(1/2)*31^(1/2)+9)^(1/3)+12*I*31^(1/2)*(3^(1/2)*31^(1/2)+9)^(1/3)+3*I*12^(1
/3)*31^(1/2)*_alpha^2*(3^(1/2)*31^(1/2)+9)^(2/3)+13*I*3^(1/2)*12^(1/3)*_alpha*(3^(1/2)*31^(1/2)+9)^(2/3)-24*(3
^(1/2)*31^(1/2)+9)^(1/3)+24*_alpha*(3^(1/2)*31^(1/2)+9)^(1/3)+11*12^(1/3)*_alpha^2*(3^(1/2)*31^(1/2)+9)^(2/3)-
13*12^(1/3)*_alpha*(3^(1/2)*31^(1/2)+9)^(2/3)-6*_alpha^2*(3^(1/2)*31^(1/2)+9)^(1/3)+4*3^(1/2)*31^(1/2)*(3^(1/2
)*31^(1/2)+9)^(1/3)))*EllipticPi(1/3*3^(1/2)*(I*(x-1/12*(108+12*93^(1/2))^(1/3)+1/(108+12*93^(1/2))^(1/3)+1/2*
I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108+12*93^(1/2))^(1/3)))*3^(1/2)/(1/6*(108+12*93^(1/2))^(1/3)+2/(10
8+12*93^(1/2))^(1/3)))^(1/2),2+1/72*_alpha*31^(1/2)*3^(1/2)*(108+12*3^(1/2)*31^(1/2))^(2/3)-1/24*_alpha^2*31^(
1/2)*3^(1/2)*(108+12*3^(1/2)*31^(1/2))^(1/3)-1/144*_alpha^2*(108+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)*3^(1/2)+1
/24*_alpha*3^(1/2)*(108+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)+5/144*I*_alpha^2*3^(1/2)*(108+12*3^(1/2)*31^(1/2))
^(2/3)-1/24*I*_alpha^2*(108+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)-1/18*I*_alpha*3^(1/2)*(108+12*3^(1/2)*31^(1/2)
)^(2/3)+7/72*I*_alpha^2*3^(1/2)*(108+12*3^(1/2)*31^(1/2))^(1/3)+1/24*I*_alpha*(108+12*3^(1/2)*31^(1/2))^(1/3)*
31^(1/2)-5/72*I*_alpha*3^(1/2)*(108+12*3^(1/2)*31^(1/2))^(1/3)+1/72*I*_alpha*(108+12*3^(1/2)*31^(1/2))^(2/3)*3
1^(1/2)-1/144*I*_alpha^2*(108+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2)-1/72*(108+12*3^(1/2)*31^(1/2))^(2/3)*31^(1/2
)*3^(1/2)+1/12*(108+12*3^(1/2)*31^(1/2))^(2/3)-1/24*I*(108+12*3^(1/2)*31^(1/2))^(1/3)*31^(1/2)-7/72*I*(108+12*
3^(1/2)*31^(1/2))^(1/3)*3^(1/2)+13/72*I*(108+12*3^(1/2)*31^(1/2))^(2/3)*3^(1/2)-1/18*I*(108+12*3^(1/2)*31^(1/2
))^(2/3)*31^(1/2)+1/3*I*31^(1/2)+13/24*(108+12*3^(1/2)*31^(1/2))^(1/3)-1/3*I*_alpha*31^(1/2)+1/6*I*_alpha^2*31
^(1/2)-13/24*_alpha*(108+12*3^(1/2)*31^(1/2))^(1/3)+1/48*(108+12*3^(1/2)*31^(1/2))^(2/3)*_alpha^2-1/12*(108+12
*3^(1/2)*31^(1/2))^(2/3)*_alpha+11/24*(108+12*3^(1/2)*31^(1/2))^(1/3)*_alpha^2+1/2*_alpha^2-1/24*(108+12*3^(1/
2)*31^(1/2))^(1/3)*31^(1/2)*3^(1/2)-2*_alpha,(I*3^(1/2)*(1/6*(108+12*93^(1/2))^(1/3)+2/(108+12*93^(1/2))^(1/3)
)/(1/4*(108+12*93^(1/2))^(1/3)-3/(108+12*93^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/6*(108+12*93^(1/2))^(1/3)-2/(108+12
*93^(1/2))^(1/3))))^(1/2)),_alpha=RootOf(_Z^3-_Z^2+_Z+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{3} - x - 2}{{\left (x^{3} - x^{2} + x + 1\right )} \sqrt {x^{3} + x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x^3 - x - 2)/((x^3 - x^2 + x + 1)*sqrt(x^3 + x + 1)), x)

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mupad [B]  time = 1.88, size = 2490, normalized size = 166.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - x^3 + 2)/((x + x^3 + 1)^(1/2)*(x - x^2 + x^3 + 1)),x)

[Out]

symsum(-(2*(-(x + 1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/((3^(
1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31
^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((x + (3^(1/2)*(1/(
3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*10
8^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)
) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3
)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1
/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*ellipticPi(((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^
(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1
/2))/108 - 1/2)^(1/3))/2)/(root(z^3 - z^2 + z + 1, z, k) + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/
3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2
)*108^(1/2))/108 - 1/2)^(1/3)/2), asin(((x + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^
(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(
1/2)), -(3^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*1i)/
(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))))*(2*root(z^3 - z^2
+ z + 1, z, k) - root(z^3 - z^2 + z + 1, z, k)^2 + 3)*((3^(1/2)*(x - (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))
+ ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*1i)/(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*
108^(1/2))/108 - 1/2)^(1/3))))^(1/2))/((x^3 - x*((1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*10
8^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2
) - (1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((
31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1
/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2) + ((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))
+ ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)/2)) - (1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*
((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6
*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2
)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/10
8 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2))^(1/2)*(3*root(z^3 - z^2 + z + 1, z, k)^2 - 2*root
(z^3 - z^2 + z + 1, z, k) + 1)*(root(z^3 - z^2 + z + 1, z, k) + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2
)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31
^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)), k, 1, 3) - (2*(-(x + 1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((
31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108
^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3))/2))^(1/2)*ellipticF(asin(((x + (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/1
08 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^
(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(
1/2)), -(3^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*1i)/
(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))))*((x + (3^(1/2)*(1
/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*
108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)/((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))
/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/
3)) + (3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(2*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (3*((31^
(1/2)*108^(1/2))/108 - 1/2)^(1/3))/2)*((3^(1/2)*(x - (3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) +
((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^
(1/2))/108 - 1/2)^(1/3)/2)*1i)/(3*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 -
1/2)^(1/3))))^(1/2))/(x^3 - x*((1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2
)^(1/3))*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)
/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2) - (1/(3*((31^(1/
2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2)
)/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1
/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2) + ((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + (
(31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(
1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108
- 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)
) - (1/(3*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) - ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*((3^(1/2)*(1/(3*((
31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 - 1/(6*((31^(1/2)*108^(1
/2))/108 - 1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2)*((3^(1/2)*(1/(3*((31^(1/2)*108^(1/2))/108 -
1/2)^(1/3)) + ((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3))*1i)/2 + 1/(6*((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)) -
((31^(1/2)*108^(1/2))/108 - 1/2)^(1/3)/2))^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {x}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\right )\, dx - \int \frac {x^{3}}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\, dx - \int \left (- \frac {2}{x^{3} \sqrt {x^{3} + x + 1} - x^{2} \sqrt {x^{3} + x + 1} + x \sqrt {x^{3} + x + 1} + \sqrt {x^{3} + x + 1}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-x/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 + x + 1)),
x) - Integral(x**3/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 + x +
1)), x) - Integral(-2/(x**3*sqrt(x**3 + x + 1) - x**2*sqrt(x**3 + x + 1) + x*sqrt(x**3 + x + 1) + sqrt(x**3 +
x + 1)), x)

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