Integrand size = 27, antiderivative size = 123 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {2 \sqrt {x+x^2+x^3}}{3 \left (1+x+x^2\right )}+\frac {1}{3} \arctan \left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{3 \sqrt {3}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.56 (sec) , antiderivative size = 785, normalized size of antiderivative = 6.38, number of steps used = 43, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2081, 6857, 730, 1117, 948, 175, 552, 551, 863, 836, 853, 1211, 1209} \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (-1,\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ),\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1+i \sqrt {3}\right ),\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\arcsin \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 {\left (1+i \sqrt {3}\right ) \sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}}+\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}} \]
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Rule 175
Rule 551
Rule 552
Rule 730
Rule 836
Rule 853
Rule 863
Rule 948
Rule 1117
Rule 1209
Rule 1211
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}}} \operatorname {EllipticPi}\left (-1,\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ),\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1+i \sqrt {3}\right ),\arcsin \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}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ),\arcsin \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*}
Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {\sqrt {x} \left (6 \sqrt {x}+3 \sqrt {1+x+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )+3 \sqrt {2} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {1+x+x^2}}\right )+\sqrt {3} \sqrt {1+x+x^2} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )\right )}{9 \sqrt {x \left (1+x+x^2\right )}} \]
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Time = 2.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{9}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right )}{3}-\frac {\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )}{3}\) | \(80\) |
default | \(-\frac {-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}}{3}+\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-2 x}{3 \sqrt {x \left (x^{2}+x +1\right )}}\) | \(111\) |
pseudoelliptic | \(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {2}}{2 x}\right ) \sqrt {x \left (x^{2}+x +1\right )}-3 \arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right ) \sqrt {x \left (x^{2}+x +1\right )}+6 x}{9 \sqrt {x \left (x^{2}+x +1\right )}}\) | \(111\) |
trager | \(\frac {2 \sqrt {x^{3}+x^{2}+x}}{3 \left (x^{2}+x +1\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}+x^{2}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}-x +1}\right )}{6}\) | \(180\) |
elliptic | \(\text {Expression too large to display}\) | \(1330\) |
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Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.59 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\frac {3 \, \sqrt {2} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 14 \, x^{3} + 4 \, \sqrt {2} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 3 \, x + 1\right )} + 19 \, x^{2} + 14 \, x + 1}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \sqrt {3} {\left (x^{2} + x + 1\right )} \log \left (\frac {x^{4} + 20 \, x^{3} + 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) - 6 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + 24 \, \sqrt {x^{3} + x^{2} + x}}{36 \, {\left (x^{2} + x + 1\right )}} \]
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\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx - \int \frac {1}{x^{6} \sqrt {x^{3} + x^{2} + x} - \sqrt {x^{3} + x^{2} + x}}\, dx \]
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\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{3} + x^{2} + x}} \,d x } \]
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\[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} \sqrt {x^{3} + x^{2} + x}} \,d x } \]
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Time = 5.50 (sec) , antiderivative size = 1195, normalized size of antiderivative = 9.72 \[ \int \frac {1+x^6}{\sqrt {x+x^2+x^3} \left (1-x^6\right )} \, dx=\text {Too large to display} \]
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