Optimal. Leaf size=123 \[ \frac {2 \sqrt {x^3+x^2+x}}{3 \left (x^2+x+1\right )}+\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )}{3 \sqrt {3}} \]
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Rubi [C] time = 2.51, antiderivative size = 785, normalized size of antiderivative = 6.38, number of steps used = 43, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {2056, 6725, 716, 1103, 934, 169, 538, 537, 849, 822, 839, 1197, 1195} \begin {gather*} \frac {2 x \left (-i \sqrt {3} x-(-1)^{2/3}+1\right )}{9 \sqrt {x^3+x^2+x}}+\frac {2 x \left (i \sqrt {3} x+\sqrt [3]{-1}+1\right )}{9 \sqrt {x^3+x^2+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x^3+x^2+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x^3+x^2+x}}-\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (-1;\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 169
Rule 537
Rule 538
Rule 716
Rule 822
Rule 839
Rule 849
Rule 934
Rule 1103
Rule 1195
Rule 1197
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1+x^6}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {2}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\right )}\right ) \, dx}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^6\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{2 \sqrt {x} \sqrt {1+x+x^2} \left (1-x^3\right )}+\frac {1}{2 \sqrt {x} \sqrt {1+x+x^2} \left (1+x^3\right )}\right ) \, dx}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2} \left (1-x^3\right )} \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2} \left (1+x^3\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt {x} \sqrt {1+x+x^2}}-\frac {1}{3 \sqrt {x} \left (-1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}}-\frac {1}{3 \sqrt {x} \left (-1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{3 (1-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {1}{3 \sqrt {x} \left (1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}}+\frac {1}{3 \sqrt {x} \left (1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(-1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+\sqrt [3]{-1} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1-(-1)^{2/3} x\right ) \sqrt {1+x+x^2}} \, dx}{3 \sqrt {x+x^2+x^3}}\\ &=-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(-1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1+\sqrt [3]{-1} x}{\sqrt {x} \left (1+x+x^2\right )^{3/2}} \, dx}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1-(-1)^{2/3} x}{\sqrt {x} \left (1+x+x^2\right )^{3/2}} \, dx}{3 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2} \left (1-\sqrt [3]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2} \left (1+(-1)^{2/3} x^2\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {\frac {1}{2} \left (2-\sqrt [3]{-1}\right )-\frac {1}{2} i \sqrt {3} x}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{9 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {\frac {1}{2} \left (2+(-1)^{2/3}\right )+\frac {1}{2} i \sqrt {3} x}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{9 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}+2 x^2} \left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}+2 x^2} \left (1+(-1)^{2/3} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (2-\sqrt [3]{-1}\right )-\frac {1}{2} i \sqrt {3} x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {x+x^2+x^3}}+\frac {\left (4 \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (2+(-1)^{2/3}\right )+\frac {1}{2} i \sqrt {3} x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{9 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\sqrt [3]{-1} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(-1)^{2/3} x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {1+x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x+x^2+x^3}}\\ &=\frac {2 x \left (1-(-1)^{2/3}-i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}+\frac {2 x \left (1+\sqrt [3]{-1}+i \sqrt {3} x\right )}{9 \sqrt {x+x^2+x^3}}-\frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x+x^2+x^3}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{6 \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (-1;\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1-i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}+\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (1+i \sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 5.02, size = 939, normalized size = 7.63 \begin {gather*} \frac {2 x \left (\sqrt {x} \left (\frac {(-1)^{2/3} \sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {\frac {(-1)^{2/3} \sqrt {x}-1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {-\frac {(-1)^{2/3} \sqrt {x}+1}{\sqrt {x}+\sqrt [3]{-1}-1}} \left (\left (1+\sqrt [3]{-1}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (-1;\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )^2}{\left (1+(-1)^{2/3}\right ) x}-\frac {(-1)^{2/3} \sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {\frac {(-1)^{2/3} \sqrt {x}-1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}} \sqrt {-\frac {(-1)^{2/3} \sqrt {x}+1}{\sqrt {x}+\sqrt [3]{-1}-1}} \left (\left (-1+\sqrt [3]{-1}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (3;\left .\sin ^{-1}\left (\sqrt {\frac {\sqrt {x}-\sqrt [3]{-1}+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )}}\right )\right |-3\right )\right ) \left (\sqrt [3]{-1} \sqrt {x}-1\right )^2}{\left (1+(-1)^{2/3}\right ) x}-2 (-1)^{2/3} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} F\left (i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\sqrt [3]{-1} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-1;i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )-\sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-1;i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+(-1)^{2/3} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\frac {3 (-1)^{2/3} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )}{\left (1+\sqrt [3]{-1}\right )^2}-\frac {3 \sqrt [3]{-1} \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )}{\left (1+\sqrt [3]{-1}\right )^2}\right )+1\right )}{3 \sqrt {x \left (x^2+x+1\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.22, size = 123, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^3+x^2+x}}{3 \left (x^2+x+1\right )}+\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 196, normalized size = 1.59 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 14 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 3 \, x + 1\right )} + 19 \, x^{2} + 14 \, x + 1}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \sqrt {3} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 20 \, x^{3} + 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - 6 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + 24 \, \sqrt {x^{3} + x^{2} + x}}{36 \, {\left (x^{2} + x + 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^{3} + x^{2} + x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 736, normalized size = 5.98
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^{3} + x^{2} + x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 1195, normalized size = 9.72
result too large to display
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 {x^{6}}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx - \int \frac {1}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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