3.11.90 \(\int \frac {(1-x^3+x^4+x^6)^{3/4} (-4+x^3+2 x^6)}{(1-x^3+x^6)^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac {\left (x^6+x^4-x^3+1\right )^{3/4} x}{x^6-x^3+1}-\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^4-x^3+1}}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^4-x^3+1}}\right ) \]

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Rubi [F]  time = 18.76, 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 {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 - x^3 + x^4 + x^6)^(3/4)*(-4 + x^3 + 2*x^6))/(1 - x^3 + x^6)^2,x]

[Out]

(-4*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 - I*Sqrt[3])^(1/3) - 2*x)^2, x])/(9*((1 - I*Sqrt[3])/2)
^(1/3)) + (16*2^(1/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 - I*Sqrt[3])^(1/3) - 2*x)^2, x])/(9*(
1 - I*Sqrt[3])^(4/3)) - (4*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 + I*Sqrt[3])^(1/3) - 2*x)^2, x])
/(9*((1 + I*Sqrt[3])/2)^(1/3)) + (16*2^(1/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 + I*Sqrt[3])^(
1/3) - 2*x)^2, x])/(9*(1 + I*Sqrt[3])^(4/3)) - (4*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 + I*Sqrt[
3])^(1/3) - 2*x), x])/(9*((1 + I*Sqrt[3])/2)^(2/3)) + (16*2^(2/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2
/3)*(1 + I*Sqrt[3])^(1/3) - 2*x), x])/(9*(1 + I*Sqrt[3])^(5/3)) + (((2*I)/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^
(3/4)/((1 - I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x])/(Sqrt[3]*(1 - I*Sqrt[3])^(2/3)) - (((2*I)/3)*Defer[Int][(1 -
x^3 + x^4 + x^6)^(3/4)/((1 + I*Sqrt[3])^(1/3) + (-2)^(1/3)*x), x])/(Sqrt[3]*(1 + I*Sqrt[3])^(2/3)) + (8*(-2)^
(1/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 - I*Sqrt[3])^(1/3) + 2*(-1)^(1/3)*x)^2, x])/(9*(1 - I
*Sqrt[3])^(1/3)) - (2*(-2)^(1/3)*(1 - I*Sqrt[3])^(2/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 - I*
Sqrt[3])^(1/3) + 2*(-1)^(1/3)*x)^2, x])/9 - (4*2^(1/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 + I*
Sqrt[3])^(1/3) + 2*(-1)^(1/3)*x)^2, x])/(9*((-I)/(I - Sqrt[3]))^(2/3)*(1 - I*Sqrt[3])) - (16*((-2*I)/(I - Sqrt
[3]))^(1/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2^(2/3)*(1 + I*Sqrt[3])^(1/3) + 2*(-1)^(1/3)*x)^2, x])/(9*
(1 - I*Sqrt[3])) - (16*(-1)^(1/6)*2^(1/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(-(2^(2/3)*(1 - I*Sqrt[3])^(1
/3)) + 2*(-1)^(2/3)*x)^2, x])/(9*(I - Sqrt[3])*(1 - I*Sqrt[3])^(1/3)) + (4*(-1)^(1/6)*2^(1/3)*(1 - I*Sqrt[3])^
(2/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(-(2^(2/3)*(1 - I*Sqrt[3])^(1/3)) + 2*(-1)^(2/3)*x)^2, x])/(9*(I
- Sqrt[3])) - (8*(-1)^(2/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(-(2^(2/3)*(1 + I*Sqrt[3])^(1/3)) + 2*(-1)^
(2/3)*x)^2, x])/(9*((1 + I*Sqrt[3])/2)^(1/3)) + (2*(-1)^(2/3)*2^(1/3)*(1 + I*Sqrt[3])^(2/3)*Defer[Int][(1 - x^
3 + x^4 + x^6)^(3/4)/(-(2^(2/3)*(1 + I*Sqrt[3])^(1/3)) + 2*(-1)^(2/3)*x)^2, x])/9 + (((2*I)/3)*Defer[Int][(1 -
x^3 + x^4 + x^6)^(3/4)/((1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x), x])/(Sqrt[3]*(1 - I*Sqrt[3])^(2/3)) - (((2*I)/3)*
Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/((1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x), x])/(Sqrt[3]*(1 + I*Sqrt[3])^(2/3)
) + (((2*I)/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/((1 - I*Sqrt[3])^(1/3) - (-1)^(2/3)*2^(1/3)*x), x])/(Sqr
t[3]*(1 - I*Sqrt[3])^(2/3)) - (((2*I)/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/((1 + I*Sqrt[3])^(1/3) - (-1)^
(2/3)*2^(1/3)*x), x])/(Sqrt[3]*(1 + I*Sqrt[3])^(2/3)) + (4*(-2)^(2/3)*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(
2^(2/3) + (1 - I*Sqrt[3])^(2/3)*x), x])/9 - (2^(2/3)*(1 + I*Sqrt[3])*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2
^(2/3) + (1 - I*Sqrt[3])^(2/3)*x), x])/9 + (4*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(3*I - Sqrt[3] + 2^(1/3)*
Sqrt[3]*(1 - I*Sqrt[3])^(2/3)*x), x])/(3*Sqrt[3]) - (4*(3*I + Sqrt[3])*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/
(3*I - Sqrt[3] + 2^(1/3)*Sqrt[3]*(1 - I*Sqrt[3])^(2/3)*x), x])/9 - (4*(1 + I*Sqrt[3])*Defer[Int][(1 - x^3 + x^
4 + x^6)^(3/4)/(2 + 2^(1/3)*(1 + I*Sqrt[3])^(2/3)*x), x])/9 + ((1 + I*Sqrt[3])^2*Defer[Int][(1 - x^3 + x^4 + x
^6)^(3/4)/(2 + 2^(1/3)*(1 + I*Sqrt[3])^(2/3)*x), x])/9 - (4*(-1)^(1/9)*(I - Sqrt[3])*Defer[Int][(1 - x^3 + x^4
+ x^6)^(3/4)/(I*2^(2/3)*(1 + I*Sqrt[3])^(1/3) + (I + Sqrt[3])*x), x])/9 - (2*(-1)^(1/9)*(I + Sqrt[3])*Defer[I
nt][(1 - x^3 + x^4 + x^6)^(3/4)/(I*2^(2/3)*(1 + I*Sqrt[3])^(1/3) + (I + Sqrt[3])*x), x])/9 + (8*Defer[Int][(1
- x^3 + x^4 + x^6)^(3/4)/(2*(3*I - Sqrt[3]) - 2^(1/3)*(1 - I*Sqrt[3])^(2/3)*(3*I + Sqrt[3])*x), x])/(3*Sqrt[3]
) - (8*(3*I + Sqrt[3])*Defer[Int][(1 - x^3 + x^4 + x^6)^(3/4)/(2*(3*I - Sqrt[3]) - 2^(1/3)*(1 - I*Sqrt[3])^(2/
3)*(3*I + Sqrt[3])*x), x])/9

Rubi steps

\begin {align*} \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx &=\int \left (\frac {3 \left (-2+x^3\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2}+\frac {2 \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}\right ) \, dx\\ &=2 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6} \, dx+3 \int \frac {\left (-2+x^3\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2} \, dx\\ &=2 \int \left (\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^3\right )}+\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx+3 \int \left (-\frac {2 \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2}+\frac {x^3 \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2}\right ) \, dx\\ &=3 \int \frac {x^3 \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2} \, dx-6 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2} \, dx+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1+i \sqrt {3}-2 x^3} \, dx}{\sqrt {3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}\\ &=3 \int \left (-\frac {2 \left (1+i \sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}-2 x^3\right )^2}+\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x^3\right )}-\frac {2 \left (1-i \sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}+2 x^3\right )^2}+\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx-6 \int \left (-\frac {4 \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}-2 x^3\right )^2}+\frac {4 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x^3\right )}-\frac {4 \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}+2 x^3\right )^2}+\frac {4 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx+\frac {(4 i) \int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}+\frac {(4 i) \int \left (\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}\\ &=8 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1+i \sqrt {3}-2 x^3\right )^2} \, dx+8 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-1+i \sqrt {3}+2 x^3\right )^2} \, dx+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1+i \sqrt {3}-2 x^3} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1+i \sqrt {3}-2 x^3} \, dx}{\sqrt {3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-1+i \sqrt {3}+2 x^3\right )^2} \, dx+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}-\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1+i \sqrt {3}-2 x^3\right )^2} \, dx\\ &=8 \int \left (-\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} \left (-i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x\right )^2}-\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3} \left (-i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x\right )}-\frac {2 i \sqrt [3]{-2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+\sqrt [3]{-1}\right )^2 \sqrt [3]{1+i \sqrt {3}} \left (-i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2}-\frac {2 \sqrt [6]{-1} \sqrt [3]{2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+(-1)^{2/3}\right )^2 \sqrt [3]{1+i \sqrt {3}} \left (-i+\sqrt {3}\right ) \left (-2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 (-1)^{2/3} x\right )^2}-\frac {i \left (-i+\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{18 \left (2+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3} x\right )}+\frac {\sqrt [9]{-1} \left (-i+\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{18 \left (i 2^{2/3} \sqrt [3]{1+i \sqrt {3}}+\left (i+\sqrt {3}\right ) x\right )}\right ) \, dx+8 \int \left (\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}-2 x\right )^2}+\frac {2 i \sqrt [3]{-2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+\sqrt [3]{-1}\right )^2 \sqrt [3]{1-i \sqrt {3}} \left (i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2}+\frac {2 \sqrt [6]{-1} \sqrt [3]{2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+(-1)^{2/3}\right )^2 \sqrt [3]{1-i \sqrt {3}} \left (i+\sqrt {3}\right ) \left (-2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 (-1)^{2/3} x\right )^2}+\frac {(-1)^{2/3} \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \sqrt [3]{2} \left (2^{2/3}+\left (1-i \sqrt {3}\right )^{2/3} x\right )}+\frac {\left (-3 i-\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{18 \left (3 i-\sqrt {3}+\sqrt [3]{2} \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3} x\right )}+\frac {\left (-3 i-\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \left (2 \left (3 i-\sqrt {3}\right )-\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3} \left (3 i+\sqrt {3}\right ) x\right )}\right ) \, dx+\frac {(2 i) \int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}+\frac {(2 i) \int \left (\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}-\frac {(8 i) \int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}-\frac {(8 i) \int \left (\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\left (2 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}-2 x\right )^2}+\frac {2 i \sqrt [3]{-2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+\sqrt [3]{-1}\right )^2 \sqrt [3]{1-i \sqrt {3}} \left (i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2}+\frac {2 \sqrt [6]{-1} \sqrt [3]{2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+(-1)^{2/3}\right )^2 \sqrt [3]{1-i \sqrt {3}} \left (i+\sqrt {3}\right ) \left (-2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 (-1)^{2/3} x\right )^2}+\frac {(-1)^{2/3} \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \sqrt [3]{2} \left (2^{2/3}+\left (1-i \sqrt {3}\right )^{2/3} x\right )}+\frac {\left (-3 i-\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{18 \left (3 i-\sqrt {3}+\sqrt [3]{2} \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3} x\right )}+\frac {\left (-3 i-\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \left (2 \left (3 i-\sqrt {3}\right )-\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3} \left (3 i+\sqrt {3}\right ) x\right )}\right ) \, dx+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}-\left (2 \left (1+i \sqrt {3}\right )\right ) \int \left (-\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} \left (-i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x\right )^2}-\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{9 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3} \left (-i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x\right )}-\frac {2 i \sqrt [3]{-2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+\sqrt [3]{-1}\right )^2 \sqrt [3]{1+i \sqrt {3}} \left (-i+\sqrt {3}\right ) \left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2}-\frac {2 \sqrt [6]{-1} \sqrt [3]{2} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+(-1)^{2/3}\right )^2 \sqrt [3]{1+i \sqrt {3}} \left (-i+\sqrt {3}\right ) \left (-2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 (-1)^{2/3} x\right )^2}-\frac {i \left (-i+\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{18 \left (2+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3} x\right )}+\frac {\sqrt [9]{-1} \left (-i+\sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{18 \left (i 2^{2/3} \sqrt [3]{1+i \sqrt {3}}+\left (i+\sqrt {3}\right ) x\right )}\right ) \, dx\\ &=\frac {1}{9} \left (4 (-2)^{2/3}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2^{2/3}+\left (1-i \sqrt {3}\right )^{2/3} x} \, dx+\frac {4 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 i-\sqrt {3}+\sqrt [3]{2} \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3} x} \, dx}{3 \sqrt {3}}+\frac {8 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2 \left (3 i-\sqrt {3}\right )-\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3} \left (3 i+\sqrt {3}\right ) x} \, dx}{3 \sqrt {3}}-\frac {1}{9} \left (4 \sqrt [9]{-1} \left (i-\sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{i 2^{2/3} \sqrt [3]{1+i \sqrt {3}}+\left (i+\sqrt {3}\right ) x} \, dx-\frac {4 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}-2 x\right )^2} \, dx}{9 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {\left (16 \sqrt [6]{-1}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 (-1)^{2/3} x\right )^2} \, dx}{9 \left (i-\sqrt {3}\right ) \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+\frac {\left (16 \sqrt [3]{2}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}-2 x\right )^2} \, dx}{9 \left (1-i \sqrt {3}\right )^{4/3}}-\frac {\left (4 \sqrt [3]{2}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2} \, dx}{9 \left (-\frac {i}{i-\sqrt {3}}\right )^{2/3} \left (1-i \sqrt {3}\right )}-\frac {\left (16 \sqrt [3]{-\frac {2 i}{i-\sqrt {3}}}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2} \, dx}{9 \left (1-i \sqrt {3}\right )}-\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (8 \sqrt [3]{-2}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2} \, dx}{9 \sqrt [3]{1-i \sqrt {3}}}-\frac {1}{9} \left (2 \sqrt [3]{-2} \left (1-i \sqrt {3}\right )^{2/3}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 \sqrt [3]{-1} x\right )^2} \, dx+\frac {\left (4 \sqrt [6]{-1} \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-2^{2/3} \sqrt [3]{1-i \sqrt {3}}+2 (-1)^{2/3} x\right )^2} \, dx}{9 \left (i-\sqrt {3}\right )}-\frac {4 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x} \, dx}{9 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {4 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x\right )^2} \, dx}{9 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {\left (8 (-1)^{2/3}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 (-1)^{2/3} x\right )^2} \, dx}{9 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {\left (16\ 2^{2/3}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x} \, dx}{9 \left (1+i \sqrt {3}\right )^{5/3}}+\frac {\left (16 \sqrt [3]{2}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (2^{2/3} \sqrt [3]{1+i \sqrt {3}}-2 x\right )^2} \, dx}{9 \left (1+i \sqrt {3}\right )^{4/3}}+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {1}{9} \left (2 (-1)^{2/3} \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-2^{2/3} \sqrt [3]{1+i \sqrt {3}}+2 (-1)^{2/3} x\right )^2} \, dx-\frac {1}{9} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3} x} \, dx-\frac {1}{9} \left (2^{2/3} \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2^{2/3}+\left (1-i \sqrt {3}\right )^{2/3} x} \, dx+\frac {1}{9} \left (1+i \sqrt {3}\right )^2 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2+\sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3} x} \, dx-\frac {1}{9} \left (2 \sqrt [9]{-1} \left (i+\sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{i 2^{2/3} \sqrt [3]{1+i \sqrt {3}}+\left (i+\sqrt {3}\right ) x} \, dx-\frac {1}{9} \left (4 \left (3 i+\sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 i-\sqrt {3}+\sqrt [3]{2} \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3} x} \, dx-\frac {1}{9} \left (8 \left (3 i+\sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{2 \left (3 i-\sqrt {3}\right )-\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3} \left (3 i+\sqrt {3}\right ) x} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 - x^3 + x^4 + x^6)^(3/4)*(-4 + x^3 + 2*x^6))/(1 - x^3 + x^6)^2,x]

[Out]

Integrate[((1 - x^3 + x^4 + x^6)^(3/4)*(-4 + x^3 + 2*x^6))/(1 - x^3 + x^6)^2, x]

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IntegrateAlgebraic [A]  time = 0.19, size = 81, normalized size = 1.00 \begin {gather*} -\frac {x \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}-\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - x^3 + x^4 + x^6)^(3/4)*(-4 + x^3 + 2*x^6))/(1 - x^3 + x^6)^2,x]

[Out]

-((x*(1 - x^3 + x^4 + x^6)^(3/4))/(1 - x^3 + x^6)) - (3*ArcTan[x/(1 - x^3 + x^4 + x^6)^(1/4)])/2 - (3*ArcTanh[
x/(1 - x^3 + x^4 + x^6)^(1/4)])/2

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fricas [B]  time = 50.53, size = 196, normalized size = 2.42 \begin {gather*} -\frac {3 \, {\left (x^{6} - x^{3} + 1\right )} \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - x^{3} + 1}\right ) - 3 \, {\left (x^{6} - x^{3} + 1\right )} \log \left (\frac {x^{6} + 2 \, x^{4} - 2 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - x^{3} + 2 \, \sqrt {x^{6} + x^{4} - x^{3} + 1} x^{2} - 2 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - x^{3} + 1}\right ) + 4 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x}{4 \, {\left (x^{6} - x^{3} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*(x^6 - x^3 + 1)*arctan(2*((x^6 + x^4 - x^3 + 1)^(1/4)*x^3 + (x^6 + x^4 - x^3 + 1)^(3/4)*x)/(x^6 - x^3
+ 1)) - 3*(x^6 - x^3 + 1)*log((x^6 + 2*x^4 - 2*(x^6 + x^4 - x^3 + 1)^(1/4)*x^3 - x^3 + 2*sqrt(x^6 + x^4 - x^3
+ 1)*x^2 - 2*(x^6 + x^4 - x^3 + 1)^(3/4)*x + 1)/(x^6 - x^3 + 1)) + 4*(x^6 + x^4 - x^3 + 1)^(3/4)*x)/(x^6 - x^3
+ 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{3} - 4\right )} {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}}}{{\left (x^{6} - x^{3} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^6 + x^3 - 4)*(x^6 + x^4 - x^3 + 1)^(3/4)/(x^6 - x^3 + 1)^2, x)

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maple [C]  time = 10.63, size = 249, normalized size = 3.07

method	result	size
trager	\(-\frac {x \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}}}{x^{6}-x^{3}+1}-\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x +2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{3}+1}\right )}{4}+\frac {3 \ln \left (-\frac {-x^{6}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-2 x^{4}+x^{3}-1}{x^{6}-x^{3}+1}\right )}{4}\)	\(249\)
risch	\(-\frac {x \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}}}{x^{6}-x^{3}+1}+\frac {3 \ln \left (-\frac {-x^{6}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-2 x^{4}+x^{3}-1}{x^{6}-x^{3}+1}\right )}{4}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{3}+1}\right )}{4}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+x^4-x^3+1)^(3/4)*(2*x^6+x^3-4)/(x^6-x^3+1)^2,x,method=_RETURNVERBOSE)

[Out]

-x*(x^6+x^4-x^3+1)^(3/4)/(x^6-x^3+1)-3/4*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)*x^6+2*RootOf(_Z^2+1)*(x^6+x^4-x^3+
1)^(1/2)*x^2-2*RootOf(_Z^2+1)*x^4+RootOf(_Z^2+1)*x^3-2*(x^6+x^4-x^3+1)^(3/4)*x+2*(x^6+x^4-x^3+1)^(1/4)*x^3-Roo
tOf(_Z^2+1))/(x^6-x^3+1))+3/4*ln(-(-x^6+2*(x^6+x^4-x^3+1)^(3/4)*x-2*(x^6+x^4-x^3+1)^(1/2)*x^2+2*(x^6+x^4-x^3+1
)^(1/4)*x^3-2*x^4+x^3-1)/(x^6-x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{3} - 4\right )} {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}}}{{\left (x^{6} - x^{3} + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^6 + x^3 - 4)*(x^6 + x^4 - x^3 + 1)^(3/4)/(x^6 - x^3 + 1)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^6+x^3-4\right )\,{\left (x^6+x^4-x^3+1\right )}^{3/4}}{{\left (x^6-x^3+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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