3.18.16 \(\int \frac {3-8 x+8 x^2-12 x^4}{x \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}} (1+2 x^2) (3-7 x+7 x^2-6 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=149 \[ \log \left (\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (\frac {1-2 x^2}{2 x^2+1}\right )^{2/3}+(1-x) \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}-2 x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}}{\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}-2 x+2}\right ) \]

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Rubi [F]  time = 7.57, 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 {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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 &=\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} \operatorname {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} \tan ^{-1}\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*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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]

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]

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IntegrateAlgebraic [A]  time = 0.38, size = 149, normalized size = 1.00 \begin {gather*} \log \left (\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (\frac {1-2 x^2}{2 x^2+1}\right )^{2/3}+(1-x) \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}-2 x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}}{\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}-2 x+2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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

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fricas [A]  time = 3.95, size = 279, normalized size = 1.87 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 5.12, size = 2146, normalized size = 14.40 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

2*ln(-(-11+33*x-114*RootOf(4*_Z^2+2*_Z+1)*x^2+28*RootOf(4*_Z^2+2*_Z+1)*x^5-84*RootOf(4*_Z^2+2*_Z+1)*x^4+42*Roo
tOf(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^4-18*(-(2*x^2-1)/(2*x^2+1))^(
2/3)-18*(-(2*x^2-1)/(2*x^2+1))^(1/3)-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+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*Ro
otOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)+98
*x^3*RootOf(4*_Z^2+2*_Z+1)+28*RootOf(4*_Z^2+2*_Z+1)^2+8*RootOf(4*_Z^2+2*_Z+1)-56*RootOf(4*_Z^2+2*_Z+1)^2*x^2-3
6*(-(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*(-(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+18*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x-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*x^3*(-(2*x^2-1)/(2*x^2+1))^(2/3))/(2*x^4-6*x
^3+7*x^2-7*x+3)/x)*RootOf(4*_Z^2+2*_Z+1)-2*RootOf(4*_Z^2+2*_Z+1)*ln((8-12*x-58*RootOf(4*_Z^2+2*_Z+1)*x^2+28*Ro
otOf(4*_Z^2+2*_Z+1)*x^5-84*RootOf(4*_Z^2+2*_Z+1)*x^4+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+15*(-(2*x^2-1)/(2*x^2+1))^(2/3)+15*(-(2*x^2-1)/(2*x^2+1))^(1/3)-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+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*Ro
otOf(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
+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)+98*x^3*RootOf(4*_Z^2+2*_Z+1)-28*RootOf(4*_Z^2+2*_Z+1)^
2-20*RootOf(4*_Z^2+2*_Z+1)+56*RootOf(4*_Z^2+2*_Z+1)^2*x^2+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*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-60*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3-15*(-(2*x^2-1)/(2*
x^2+1))^(2/3)*x-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*x^3*(-(2*x^2-1)/(2*x^2+1))^(2/3))/(2*x^4-6*x^3+7*x^2-7*x+3)/x)-ln((8-12*x-58*RootOf(4*_Z^2+2*_Z
+1)*x^2+28*RootOf(4*_Z^2+2*_Z+1)*x^5-84*RootOf(4*_Z^2+2*_Z+1)*x^4+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+15*(-(2*x^2-1)/(2*x^2+1))^(2/3)+15*(-(2*x^2-1)/(2*x^2+1))^(1/3)-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+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+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)+98*x^3*RootOf(4*_Z^2+2*_Z+1)-28*RootOf(4*
_Z^2+2*_Z+1)^2-20*RootOf(4*_Z^2+2*_Z+1)+56*RootOf(4*_Z^2+2*_Z+1)^2*x^2+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*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-60*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3-15*(-
(2*x^2-1)/(2*x^2+1))^(2/3)*x-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*x^3*(-(2*x^2-1)/(2*x^2+1))^(2/3))/(2*x^4-6*x^3+7*x^2-7*x+3)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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((-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

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