Optimal. Leaf size=138 \[ \log \left (\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (\frac {1-2 x^2}{2 x^2+1}\right )^{2/3}+(1-x) \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}-2 x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x-\sqrt {3}}{-2 \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}+x-1}\right ) \]
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Rubi [F] time = 10.10, 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 {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \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 {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx &=\frac {\left (1-2 x^2\right )^{2/3} \int \frac {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {\left (1-2 x^2\right )^{2/3} \int \left (\frac {6}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}}+\frac {1}{x \left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}}+\frac {-22+51 x-44 x^2+22 x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}\right ) \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {\left (1-2 x^2\right )^{2/3} \int \frac {1}{x \left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (1-2 x^2\right )^{2/3} \int \frac {-22+51 x-44 x^2+22 x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (6 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {6 x \left (1-2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {1}{3};\frac {3}{2};2 x^2,-2 x^2\right )}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (1-2 x^2\right )^{2/3} \operatorname {Subst}\left (\int \frac {1}{(1-2 x)^{2/3} x \sqrt [3]{1+2 x}} \, dx,x,x^2\right )}{2 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (1-2 x^2\right )^{2/3} \int \left (-\frac {22}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}+\frac {51 x}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}-\frac {44 x^2}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}+\frac {22 x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}\right ) \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {6 x \left (1-2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {1}{3};\frac {3}{2};2 x^2,-2 x^2\right )}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\sqrt {3} \left (1-2 x^2\right )^{2/3} \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 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}-\frac {\left (1-2 x^2\right )^{2/3} \log (x)}{2 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {3 \left (1-2 x^2\right )^{2/3} \log \left (\sqrt [3]{1-2 x^2}-\sqrt [3]{1+2 x^2}\right )}{4 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}-\frac {\left (22 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (22 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}-\frac {\left (44 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (51 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \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 = 3.09, size = 138, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {-\sqrt {3}+\sqrt {3} x}{-1+x-2 \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 [B] time = 3.61, size = 278, normalized size = 2.01 \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 {{\left (12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3\right )} {\left (x - 1\right )}}{{\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 {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.30, size = 1426, normalized size = 10.33
method | result | size |
trager | \(\ln \left (\frac {14-21 x -15 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}+15 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-18 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+36 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-14 x^{5}+21 x^{2}-49 x^{3}+42 x^{4}+58 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+44 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}-12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}-60 x^{4} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+120 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-90 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+60 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+6 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-88 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-44 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}+132 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-154 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+38 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-66 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-54 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{2}+30 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}-36 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{4}+72 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{3}-30 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}}{x \left (2 x^{4}-6 x^{3}+7 x^{2}-7 x +3\right )}\right )+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {7-21 x +18 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-18 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}+15 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-30 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-14 x^{5}+35 x^{2}-49 x^{3}+42 x^{4}-22 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}-12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}+72 x^{4} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-144 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+108 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-72 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+6 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-32 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-48 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}+56 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}-12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+24 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +36 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}+45 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{2}-36 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}+30 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{4}-60 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{3}+36 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}}{x \left (2 x^{4}-6 x^{3}+7 x^{2}-7 x +3\right )}\right )\) | \(1426\) |
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 {{\left (12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3\right )} {\left (x - 1\right )}}{{\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 {2}{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 {\left (x-1\right )\,\left (12\,x^4-8\,x^2+8\,x-3\right )}{x\,\left (2\,x^2+1\right )\,{\left (-\frac {2\,x^2-1}{2\,x^2+1}\right )}^{2/3}\,\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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