Optimal. Leaf size=53 \[ \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4-x^2+1}}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right ) \]
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Rubi [C] time = 2.15, antiderivative size = 228, normalized size of antiderivative = 4.30, number of steps used = 95, number of rules used = 19, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {1586, 6725, 1728, 1208, 1139, 1103, 1195, 1210, 1698, 206, 1247, 734, 843, 619, 215, 724, 1197, 1216, 1706} \begin {gather*} \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4-x^2+1}}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right )+\frac {\left (1+(-1)^{2/3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4-x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {x^4-x^2+1}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4-x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {x^4-x^2+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4-x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {x^4-x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 619
Rule 724
Rule 734
Rule 843
Rule 1103
Rule 1139
Rule 1195
Rule 1197
Rule 1208
Rule 1210
Rule 1216
Rule 1247
Rule 1586
Rule 1698
Rule 1706
Rule 1728
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^6}{\sqrt {1-x^2+x^4} \left (1-x^6\right )} \, dx &=\int \frac {\left (1+x^2\right ) \sqrt {1-x^2+x^4}}{1-x^6} \, dx\\ &=\int \left (\frac {\sqrt {1-x^2+x^4}}{3 (1-x)}+\frac {\sqrt {1-x^2+x^4}}{3 (1+x)}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}{6 \left (1-\sqrt [3]{-1} x\right )}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}{6 \left (1+\sqrt [3]{-1} x\right )}+\frac {\left (1+(-1)^{2/3}\right ) \sqrt {1-x^2+x^4}}{6 \left (1-(-1)^{2/3} x\right )}+\frac {\left (1+(-1)^{2/3}\right ) \sqrt {1-x^2+x^4}}{6 \left (1+(-1)^{2/3} x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {\sqrt {1-x^2+x^4}}{1-x} \, dx+\frac {1}{3} \int \frac {\sqrt {1-x^2+x^4}}{1+x} \, dx+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1-\sqrt [3]{-1} x} \, dx+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1+\sqrt [3]{-1} x} \, dx+\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1-(-1)^{2/3} x} \, dx+\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1+(-1)^{2/3} x} \, dx\\ &=2 \left (\frac {1}{3} \int \frac {\sqrt {1-x^2+x^4}}{1-x^2} \, dx\right )+2 \left (\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1-(-1)^{2/3} x^2} \, dx\right )+2 \left (\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {\sqrt {1-x^2+x^4}}{1+\sqrt [3]{-1} x^2} \, dx\right )\\ &=2 \left (-\left (\frac {1}{3} \int \frac {x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+\frac {1}{3} \int \frac {1}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx\right )+2 \left (\frac {1}{3} \int \frac {1}{\left (1+\sqrt [3]{-1} x^2\right ) \sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \left (-1+\sqrt [3]{-1}\right ) \int \frac {1+\sqrt [3]{-1}-\sqrt [3]{-1} x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+2 \left (\frac {1}{3} \int \frac {1}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {1-(-1)^{2/3}+(-1)^{2/3} x^2}{\sqrt {1-x^2+x^4}} \, dx\right )\\ &=2 \left (\frac {1}{6} \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1-x^2+x^4}} \, dx-\frac {1}{3} \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{3} \int \frac {1-x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+\frac {\int \frac {1}{\sqrt {1-x^2+x^4}} \, dx}{3 \left (1-\sqrt [3]{-1}\right )}+\frac {1}{6} \left (-1+\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx-\frac {\left (\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right )\right ) \int \frac {1+x^2}{\left (1+\sqrt [3]{-1} x^2\right ) \sqrt {1-x^2+x^4}} \, dx}{3 \left (1-(-1)^{2/3}\right )}\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1-x^2+x^4}} \, dx\right )+\frac {1}{3} \left (1-\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{6} \left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {1}{\sqrt {1-x^2+x^4}} \, dx+\frac {1}{3} \left (1+(-1)^{2/3}\right ) \int \frac {1+x^2}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1-x^2+x^4}} \, dx\right )\\ &=2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )+2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}-\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )+2 \left (-\frac {x \sqrt {1-x^2+x^4}}{3 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{3 \sqrt {1-x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1-x^2+x^4}}\right )\right )\\ &=2 \left (-\frac {x \sqrt {1-x^2+x^4}}{3 \left (1+x^2\right )}+\frac {1}{6} \tanh ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{3 \sqrt {1-x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}\right )+2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \left (1-\sqrt [3]{-1}\right ) \sqrt {1-x^2+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )+2 \left (\frac {x \sqrt {1-x^2+x^4}}{6 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2+x^4}}\right )}{6 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{6 \sqrt {1-x^2+x^4}}-\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {3}{4}\right )}{12 \sqrt {1-x^2+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {3}{4}\right )}{4 \sqrt {3} \sqrt {1-x^2+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 1.80, size = 542, normalized size = 10.23 \begin {gather*} \frac {9 \sqrt [6]{-1} \sqrt {1-\sqrt [3]{-1} x^2} \sqrt {(-1)^{2/3} x^2+1} F\left (i \sinh ^{-1}\left (\sqrt [3]{-1} x\right )|(-1)^{2/3}\right )-\frac {18 i \sqrt {1-\sqrt [3]{-1} x^2} \sqrt {(-1)^{2/3} x^2+1} \Pi \left (-1;i \sinh ^{-1}\left (\sqrt [3]{-1} x\right )|(-1)^{2/3}\right )}{\left (1+\sqrt [3]{-1}\right )^2}-6 \sqrt [6]{-1} \sqrt {1-\sqrt [3]{-1} x^2} \sqrt {(-1)^{2/3} x^2+1} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left (\sqrt [3]{-1} x\right )|(-1)^{2/3}\right )-4 \sqrt [3]{-1} \sqrt {3} \left (1+(-1)^{2/3}\right ) \sqrt {\frac {x+(-1)^{5/6}}{\left ((-1)^{2/3}-1\right ) \left (\sqrt [6]{-1}-x\right )}} \sqrt {\frac {2 i x+i \sqrt {3}+1}{-\sqrt {3} x+i x+2}} \sqrt {\frac {\left (\sqrt {3}+3 i\right ) x+2 i \sqrt {3}}{-2+\left (\sqrt {3}-i\right ) x}} \left (\sqrt [6]{-1}-x\right )^2 \left (\sqrt [6]{-1} \left (\Pi \left (-\frac {2 \left (1+(-1)^{2/3}\right )}{(3+6 i)+(2+3 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (3 i+\sqrt {3}\right ) x+2 i \sqrt {3}}{\left (-i+\sqrt {3}\right ) x-2}}\right )|-\frac {1}{3}\right )-\Pi \left (\frac {2 i \left (1+(-1)^{2/3}\right )}{(-6-3 i)+(3+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (3 i+\sqrt {3}\right ) x+2 i \sqrt {3}}{\left (-i+\sqrt {3}\right ) x-2}}\right )|-\frac {1}{3}\right )\right )-F\left (\sin ^{-1}\left (\sqrt {\frac {\left (3 i+\sqrt {3}\right ) x+2 i \sqrt {3}}{\left (-i+\sqrt {3}\right ) x-2}}\right )|-\frac {1}{3}\right )\right )}{9 \sqrt {x^4-x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.30, size = 53, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4-x^2+1}}\right )+\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 62, normalized size = 1.17 \begin {gather*} \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} - x^{2} + 1} x}{x^{4} - 3 \, x^{2} + 1}\right ) + \frac {1}{3} \, \log \left (\frac {x + \sqrt {x^{4} - x^{2} + 1}}{x^{2} - 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 {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{4} - x^{2} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 944, normalized size = 17.81
result too large to display
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 {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{4} - x^{2} + 1}}\,{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 = -0.02 \begin {gather*} \int -\frac {x^6+1}{\left (x^6-1\right )\,\sqrt {x^4-x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {x^{4} - x^{2} + 1}}{x^{6} - 1}\, dx - \int \frac {x^{2} \sqrt {x^{4} - x^{2} + 1}}{x^{6} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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