Optimal. Leaf size=85 \[ -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}} \]
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Rubi [C] time = 3.07, antiderivative size = 420, normalized size of antiderivative = 4.94, number of steps used = 171, number of rules used = 18, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {1586, 6725, 1729, 1209, 1198, 220, 1196, 1211, 1699, 206, 1248, 735, 844, 215, 725, 203, 1217, 1707} \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}+\frac {\left (3+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}}-\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {x^4+1}}+\frac {\left (3-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}}-\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {x^4+1}}-\frac {\left (-\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (-\sqrt {3}+3 i\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 215
Rule 220
Rule 725
Rule 735
Rule 844
Rule 1196
Rule 1198
Rule 1209
Rule 1211
Rule 1217
Rule 1248
Rule 1586
Rule 1699
Rule 1707
Rule 1729
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx &=\int \frac {\sqrt {1+x^4} \left (1-x^4+x^8\right )}{-1+x^{12}} \, dx\\ &=\int \left (-\frac {\sqrt {1+x^4}}{12 (1-x)}-\frac {\sqrt {1+x^4}}{12 (1-i x)}-\frac {\sqrt {1+x^4}}{12 (1+i x)}-\frac {\sqrt {1+x^4}}{12 (1+x)}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-\sqrt [6]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+\sqrt [6]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-(-1)^{2/3} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+(-1)^{2/3} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1-(-1)^{5/6} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {1+x^4}}{12 \left (1+(-1)^{5/6} x\right )}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1-x} \, dx\right )-\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1-i x} \, dx-\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1+i x} \, dx-\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1+x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-\sqrt [6]{-1} x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+\sqrt [6]{-1} x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-(-1)^{2/3} x} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+(-1)^{2/3} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-\sqrt [3]{-1} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+\sqrt [3]{-1} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-(-1)^{5/6} x} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+(-1)^{5/6} x} \, dx\\ &=-2 \left (\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1-x^2} \, dx\right )-2 \left (\frac {1}{12} \int \frac {\sqrt {1+x^4}}{1+x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-\sqrt [3]{-1} x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+\sqrt [3]{-1} x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-(-1)^{2/3} x^2} \, dx\right )+2 \left (\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+(-1)^{2/3} x^2} \, dx\right )\\ &=-2 \left (-\left (\frac {1}{12} \int \frac {1+x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{6} \int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx\right )-2 \left (-\left (\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{6} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1+\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1-i \sqrt {3}\right ) \int \frac {1-\sqrt [3]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1-i \sqrt {3}\right ) \int \frac {1+\sqrt [3]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1+i \sqrt {3}\right ) \int \frac {1-(-1)^{2/3} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{12} \left (1+i \sqrt {3}\right ) \int \frac {1+(-1)^{2/3} x^2}{\sqrt {1+x^4}} \, dx\right )\\ &=-2 \left (\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {1}{12} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{12} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{6} \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {\int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (1-\sqrt [3]{-1}\right )}+\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^4}} \, dx}{6 \left (1-(-1)^{2/3}\right )}+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {\int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}-\frac {\sqrt [3]{-1} \int \frac {1+x^2}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}+\frac {1}{12} \left (3-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (1+(-1)^{2/3}\right ) \int \frac {1+x^2}{\left (1-(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {\int \frac {1+x^2}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}-\frac {\left (1+(-1)^{2/3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (1+\sqrt [3]{-1}\right )}+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )\\ &=2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1-\sqrt [3]{-1}\right ) \sqrt {1+x^4}}-\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+(-1)^{2/3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\right )-2 \left (\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\right )\\ &=-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{12 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}\right )-2 \left (\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{12 \sqrt {2}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}\right )+2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1-\sqrt [3]{-1}\right ) \sqrt {1+x^4}}-\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+(-1)^{2/3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {1}{12} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}+\frac {\left (3-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.92, size = 204, normalized size = 2.40 \begin {gather*} \frac {(-1)^{5/12} \left (-3 F\left (\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (i;\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )+\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {(1+i) x}{\sqrt {2}}\right )\right |-1\right )\right )}{\sqrt {3} \left (1+\sqrt [3]{-1}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.42, size = 85, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}-\frac {1}{3} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 120, normalized size = 1.41 \begin {gather*} -\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + \frac {1}{24} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 1} x}{x^{4} - x^{2} + 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 1} x + 1}{x^{4} - 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^{12} + 1}{{\left (x^{12} - 1\right )} \sqrt {x^{4} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 730, normalized size = 8.59
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^{12} + 1}{{\left (x^{12} - 1\right )} \sqrt {x^{4} + 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 = -0.01 \begin {gather*} \int \frac {x^{12}+1}{\sqrt {x^4+1}\,\left (x^{12}-1\right )} \,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} + 1} \left (x^{8} - x^{4} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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