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.
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\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.
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IntegrateAlgebraic [A] time = 0.40, size = 149, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\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 ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 4.04, 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.
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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.
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maple [C] time = 5.37, size = 2145, normalized size = 14.40
method | result | size |
trager | \(\text {Expression too large to display}\) | \(2145\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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