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]
[Out]
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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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