\(\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} (1+2 x^2) (3-7 x+7 x^2-6 x^3+2 x^4)} \, dx\) [2064]
Optimal result
Integrand size = 71, antiderivative size = 149 \[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}{2-2 x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}\right )+\log \left (-1+x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}+\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3}\right )
\]
[Out]
3^(1/2)*arctan(3^(1/2)*((-2*x^2+1)/(2*x^2+1))^(1/3)/(2-2*x+((-2*x^2+1)/(2*x^2+1))^(1/3)))+ln(-1+x+((-2*x^2+1)/
(2*x^2+1))^(1/3))-1/2*ln(1-2*x+x^2+(1-x)*((-2*x^2+1)/(2*x^2+1))^(1/3)+((-2*x^2+1)/(2*x^2+1))^(2/3))
Rubi [F]
\[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx
\]
[In]
Int[(3 - 8*x + 8*x^2 - 12*x^4)/(x*((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x^2)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4
)),x]
[Out]
(Sqrt[3]*(1 - 2*x^2)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 - 2*x^2)^(1/3))/(Sqrt[3]*(1 + 2*x^2)^(1/3))])/(2*((1 - 2*x
^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x^2)^(1/3)) - ((1 - 2*x^2)^(1/3)*Log[x])/(2*((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 +
2*x^2)^(1/3)) + (3*(1 - 2*x^2)^(1/3)*Log[(1 - 2*x^2)^(1/3) - (1 + 2*x^2)^(1/3)])/(4*((1 - 2*x^2)/(1 + 2*x^2))
^(1/3)*(1 + 2*x^2)^(1/3)) + ((1 - 2*x^2)^(1/3)*Defer[Int][1/((1 - 2*x^2)^(1/3)*(1 + 2*x^2)^(2/3)*(-3 + 7*x - 7
*x^2 + 6*x^3 - 2*x^4)), x])/(((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x^2)^(1/3)) + ((1 - 2*x^2)^(1/3)*Defer[Int
][x/((1 - 2*x^2)^(1/3)*(1 + 2*x^2)^(2/3)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)), x])/(((1 - 2*x^2)/(1 + 2*x^2))^(1
/3)*(1 + 2*x^2)^(1/3)) + (6*(1 - 2*x^2)^(1/3)*Defer[Int][x^2/((1 - 2*x^2)^(1/3)*(1 + 2*x^2)^(2/3)*(3 - 7*x + 7
*x^2 - 6*x^3 + 2*x^4)), x])/(((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x^2)^(1/3)) - (14*(1 - 2*x^2)^(1/3)*Defer[
Int][x^3/((1 - 2*x^2)^(1/3)*(1 + 2*x^2)^(2/3)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)), x])/(((1 - 2*x^2)/(1 + 2*x^2
))^(1/3)*(1 + 2*x^2)^(1/3))
Rubi steps \begin{align*}
\text {integral}& = \frac {\sqrt [3]{1-2 x^2} \int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}} \\ & = \frac {\sqrt [3]{1-2 x^2} \int \left (\frac {1}{x \sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3}}+\frac {-1+x+6 x^2-14 x^3}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}\right ) \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}} \\ & = \frac {\sqrt [3]{1-2 x^2} \int \frac {1}{x \sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3}} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}+\frac {\sqrt [3]{1-2 x^2} \int \frac {-1+x+6 x^2-14 x^3}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}} \\ & = \frac {\sqrt [3]{1-2 x^2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-2 x} x (1+2 x)^{2/3}} \, dx,x,x^2\right )}{2 \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}+\frac {\sqrt [3]{1-2 x^2} \int \left (\frac {1}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (-3+7 x-7 x^2+6 x^3-2 x^4\right )}+\frac {x}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}+\frac {6 x^2}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}-\frac {14 x^3}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}\right ) \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}} \\ & = \frac {\sqrt {3} \sqrt [3]{1-2 x^2} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-2 x^2}}{\sqrt {3} \sqrt [3]{1+2 x^2}}\right )}{2 \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}-\frac {\sqrt [3]{1-2 x^2} \log (x)}{2 \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}+\frac {3 \sqrt [3]{1-2 x^2} \log \left (\sqrt [3]{1-2 x^2}-\sqrt [3]{1+2 x^2}\right )}{4 \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}+\frac {\sqrt [3]{1-2 x^2} \int \frac {1}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (-3+7 x-7 x^2+6 x^3-2 x^4\right )} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}+\frac {\sqrt [3]{1-2 x^2} \int \frac {x}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}+\frac {\left (6 \sqrt [3]{1-2 x^2}\right ) \int \frac {x^2}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}}-\frac {\left (14 \sqrt [3]{1-2 x^2}\right ) \int \frac {x^3}{\sqrt [3]{1-2 x^2} \left (1+2 x^2\right )^{2/3} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \sqrt [3]{1+2 x^2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 10.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00
\[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}{2-2 x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}\right )+\log \left (-1+x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}+\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3}\right )
\]
[In]
Integrate[(3 - 8*x + 8*x^2 - 12*x^4)/(x*((1 - 2*x^2)/(1 + 2*x^2))^(1/3)*(1 + 2*x^2)*(3 - 7*x + 7*x^2 - 6*x^3 +
2*x^4)),x]
[Out]
Sqrt[3]*ArcTan[(Sqrt[3]*((1 - 2*x^2)/(1 + 2*x^2))^(1/3))/(2 - 2*x + ((1 - 2*x^2)/(1 + 2*x^2))^(1/3))] + Log[-1
+ x + ((1 - 2*x^2)/(1 + 2*x^2))^(1/3)] - Log[1 - 2*x + x^2 + (1 - x)*((1 - 2*x^2)/(1 + 2*x^2))^(1/3) + ((1 -
2*x^2)/(1 + 2*x^2))^(2/3)]/2
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.90 (sec) , antiderivative size = 2146, normalized size of antiderivative =
14.40
| | |
| method | result | size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(2146\) |
| | |
|
|
|
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[In]
int((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x,method=_RETURNV
ERBOSE)
[Out]
-2*RootOf(4*_Z^2+2*_Z+1)*ln((8-12*x-28*RootOf(4*_Z^2+2*_Z+1)^2-20*RootOf(4*_Z^2+2*_Z+1)+15*(-(2*x^2-1)/(2*x^2+
1))^(2/3)+15*(-(2*x^2-1)/(2*x^2+1))^(1/3)-84*RootOf(4*_Z^2+2*_Z+1)*x^4+56*RootOf(4*_Z^2+2*_Z+1)^2*x^2+98*RootO
f(4*_Z^2+2*_Z+1)*x^3-58*RootOf(4*_Z^2+2*_Z+1)*x^2+42*RootOf(4*_Z^2+2*_Z+1)*x-30*x*(-(2*x^2-1)/(2*x^2+1))^(1/3)
-8*x^5+12*x^2-28*x^3+24*x^4+28*RootOf(4*_Z^2+2*_Z+1)*x^5+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)
*x^3-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))
^(2/3)*x^2+24*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3-18*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x
^2+1))^(1/3)*x^2+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)
/(2*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*
x^2+1))^(1/3)+30*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-60*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3+30*(-(2*x^2-1)/(2*x^2+1)
)^(2/3)*x^2+45*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2-30*x^3*(-(2*x^2-1)/(2*x^2+1))^(2/3)-15*(-(2*x^2-1)/(2*x^2+1))^
(2/3)*x)/x/(2*x^4-6*x^3+7*x^2-7*x+3))+2*RootOf(4*_Z^2+2*_Z+1)*ln(-(-11+33*x+28*RootOf(4*_Z^2+2*_Z+1)^2+8*RootO
f(4*_Z^2+2*_Z+1)-18*(-(2*x^2-1)/(2*x^2+1))^(2/3)-18*(-(2*x^2-1)/(2*x^2+1))^(1/3)-84*RootOf(4*_Z^2+2*_Z+1)*x^4-
56*RootOf(4*_Z^2+2*_Z+1)^2*x^2+98*RootOf(4*_Z^2+2*_Z+1)*x^3-114*RootOf(4*_Z^2+2*_Z+1)*x^2+42*RootOf(4*_Z^2+2*_
Z+1)*x+36*x*(-(2*x^2-1)/(2*x^2+1))^(1/3)+22*x^5-55*x^2+77*x^3-66*x^4+28*RootOf(4*_Z^2+2*_Z+1)*x^5+12*RootOf(4*
_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^3-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-12*Roo
tOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2+24*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3-
18*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3
)*x+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3
)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)-36*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4+72*(-(2*x^2-1)/(2*x
^2+1))^(1/3)*x^3-36*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2-54*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2+36*x^3*(-(2*x^2-1)/(2
*x^2+1))^(2/3)+18*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x)/x/(2*x^4-6*x^3+7*x^2-7*x+3))-ln((8-12*x-28*RootOf(4*_Z^2+2*_
Z+1)^2-20*RootOf(4*_Z^2+2*_Z+1)+15*(-(2*x^2-1)/(2*x^2+1))^(2/3)+15*(-(2*x^2-1)/(2*x^2+1))^(1/3)-84*RootOf(4*_Z
^2+2*_Z+1)*x^4+56*RootOf(4*_Z^2+2*_Z+1)^2*x^2+98*RootOf(4*_Z^2+2*_Z+1)*x^3-58*RootOf(4*_Z^2+2*_Z+1)*x^2+42*Roo
tOf(4*_Z^2+2*_Z+1)*x-30*x*(-(2*x^2-1)/(2*x^2+1))^(1/3)-8*x^5+12*x^2-28*x^3+24*x^4+28*RootOf(4*_Z^2+2*_Z+1)*x^5
+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^3-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/
3)*x^4-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2+24*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1
))^(1/3)*x^3-18*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2
*x^2+1))^(1/3)*x+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2
*x^2+1))^(2/3)-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)+30*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-60*(-(
2*x^2-1)/(2*x^2+1))^(1/3)*x^3+30*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2+45*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2-30*x^3*(
-(2*x^2-1)/(2*x^2+1))^(2/3)-15*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x)/x/(2*x^4-6*x^3+7*x^2-7*x+3))
Fricas [A] (verification not implemented)
none
Time = 2.53 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.87
\[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=-\sqrt {3} \arctan \left (\frac {434 \, \sqrt {3} {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}} + 682 \, \sqrt {3} {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}} + \sqrt {3} {\left (242 \, x^{5} - 726 \, x^{4} + 847 \, x^{3} - 1095 \, x^{2} + 363 \, x + 124\right )}}{2662 \, x^{5} - 7986 \, x^{4} + 9317 \, x^{3} - 5969 \, x^{2} + 3993 \, x - 1674}\right ) + \frac {1}{2} \, \log \left (\frac {2 \, x^{5} - 6 \, x^{4} + 7 \, x^{3} - 7 \, x^{2} + 3 \, {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}} + 3 \, {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}} + 3 \, x}{2 \, x^{5} - 6 \, x^{4} + 7 \, x^{3} - 7 \, x^{2} + 3 \, x}\right )
\]
[In]
integrate((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x, algorith
m="fricas")
[Out]
-sqrt(3)*arctan((434*sqrt(3)*(2*x^3 - 2*x^2 + x - 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(2/3) + 682*sqrt(3)*(2*x^4 - 4
*x^3 + 3*x^2 - 2*x + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/3) + sqrt(3)*(242*x^5 - 726*x^4 + 847*x^3 - 1095*x^2 + 3
63*x + 124))/(2662*x^5 - 7986*x^4 + 9317*x^3 - 5969*x^2 + 3993*x - 1674)) + 1/2*log((2*x^5 - 6*x^4 + 7*x^3 - 7
*x^2 + 3*(2*x^3 - 2*x^2 + x - 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(2/3) + 3*(2*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*(-(2*x
^2 - 1)/(2*x^2 + 1))^(1/3) + 3*x)/(2*x^5 - 6*x^4 + 7*x^3 - 7*x^2 + 3*x))
Sympy [F(-1)]
Timed out. \[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\text {Timed out}
\]
[In]
integrate((-12*x**4+8*x**2-8*x+3)/x/((-2*x**2+1)/(2*x**2+1))**(1/3)/(2*x**2+1)/(2*x**4-6*x**3+7*x**2-7*x+3),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\int { -\frac {12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3}{{\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2} - 7 \, x + 3\right )} {\left (2 \, x^{2} + 1\right )} x \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}}} \,d x }
\]
[In]
integrate((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x, algorith
m="maxima")
[Out]
-integrate((12*x^4 - 8*x^2 + 8*x - 3)/((2*x^4 - 6*x^3 + 7*x^2 - 7*x + 3)*(2*x^2 + 1)*x*(-(2*x^2 - 1)/(2*x^2 +
1))^(1/3)), x)
Giac [F]
\[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=\int { -\frac {12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3}{{\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2} - 7 \, x + 3\right )} {\left (2 \, x^{2} + 1\right )} x \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}}} \,d x }
\]
[In]
integrate((-12*x^4+8*x^2-8*x+3)/x/((-2*x^2+1)/(2*x^2+1))^(1/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x, algorith
m="giac")
[Out]
integrate(-(12*x^4 - 8*x^2 + 8*x - 3)/((2*x^4 - 6*x^3 + 7*x^2 - 7*x + 3)*(2*x^2 + 1)*x*(-(2*x^2 - 1)/(2*x^2 +
1))^(1/3)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx=-\int \frac {12\,x^4-8\,x^2+8\,x-3}{x\,\left (2\,x^2+1\right )\,{\left (-\frac {2\,x^2-1}{2\,x^2+1}\right )}^{1/3}\,\left (2\,x^4-6\,x^3+7\,x^2-7\,x+3\right )} \,d x
\]
[In]
int(-(8*x - 8*x^2 + 12*x^4 - 3)/(x*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/3)*(7*x^2 - 7*x - 6*x^3 + 2*x^4 +
3)),x)
[Out]
-int((8*x - 8*x^2 + 12*x^4 - 3)/(x*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/3)*(7*x^2 - 7*x - 6*x^3 + 2*x^4 +
3)), x)