Optimal. Leaf size=89 \[ -\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right )}{4 \sqrt {3}} \]
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Rubi [C] time = 2.23, antiderivative size = 592, normalized size of antiderivative = 6.65, number of steps used = 31, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2056, 6725, 716, 1103, 934, 169, 538, 537} \begin {gather*} \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 {3 \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 (-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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {\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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {\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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {3 \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 (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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 169
Rule 537
Rule 538
Rule 716
Rule 934
Rule 1103
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1-x^2+x^4}{\sqrt {x+x^2+x^3} \left (-1+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1-x^2+x^4}{\sqrt {x} \sqrt {1+x+x^2} \left (-1+x^4\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-x^2}{\sqrt {x} \sqrt {1+x+x^2} \left (-1+x^4\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 (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {2-x^2}{\sqrt {x} \sqrt {1+x+x^2} \left (-1+x^4\right )} \, dx}{\sqrt {x+x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{2 \sqrt {x} \left (1-x^2\right ) \sqrt {1+x+x^2}}-\frac {3}{2 \sqrt {x} \left (1+x^2\right ) \sqrt {1+x+x^2}}\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} \left (1-x^2\right ) \sqrt {1+x+x^2}} \, dx}{2 \sqrt {x+x^2+x^3}}-\frac {\left (3 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right ) \sqrt {1+x+x^2}} \, dx}{2 \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}{2 (1-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {1}{2 \sqrt {x} (1+x) \sqrt {1+x+x^2}}\right ) \, dx}{2 \sqrt {x+x^2+x^3}}-\frac {\left (3 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {i}{2 (i-x) \sqrt {x} \sqrt {1+x+x^2}}+\frac {i}{2 \sqrt {x} (i+x) \sqrt {1+x+x^2}}\right ) \, dx}{2 \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 (3 i \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (3 i \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {1+x+x^2}} \, dx}{4 \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}{4 \sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {1+x+x^2}} \, dx}{4 \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 (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \sqrt {x+x^2+x^3}}-\frac {\left (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \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}{4 \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} (1+x) \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{4 \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 (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (3 i \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{2 \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 ) \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 )}{2 \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 ) \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 )}{2 \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 (3 i \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 (-i-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 )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (3 i \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 (-i+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 )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\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 )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\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 )}{2 \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 (3 i \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 (-i-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 )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (3 i \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 (-i+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 )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\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 )}{2 \sqrt {x+x^2+x^3}}+\frac {\left (\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 )}{2 \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 {3 \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 (-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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {\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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {\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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {3 \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 (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 )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 3.07, size = 774, normalized size = 8.70 \begin {gather*} -\frac {(-1)^{2/3} x^{3/2} \left (-4 \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 {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 )+3 \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+3 \sqrt {1-\frac {(-1)^{2/3}}{x}} \sqrt {\frac {x+\sqrt [3]{-1}}{x}} \Pi \left ((-1)^{5/6};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )-\frac {\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 (\sqrt [3]{-1} \sqrt {x}-1\right )^2 \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}-1\right )^2 x}+\frac {\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 (\sqrt [3]{-1} \sqrt {x}-1\right )^2 \left (\left (\sqrt [3]{-1}-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}-1\right )^2 x}\right )}{2 \sqrt {x \left (x^2+x+1\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.16, size = 89, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 120, normalized size = 1.35 \begin {gather*} \frac {1}{48} \, \sqrt {3} \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 ) + \frac {1}{8} \, \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) + \frac {3}{4} \, \log \left (\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{3} + x^{2} + x} + 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^{4} - x^{2} + 1}{{\left (x^{4} - 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 = 0.84, size = 136, normalized size = 1.53
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +\RootOf \left (\textit {\_Z}^{2}-3\right )-6 \sqrt {x^{3}+x^{2}+x}}{\left (-1+x \right )^{2}}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{8}+\frac {3 \ln \left (\frac {-x^{2}+2 \sqrt {x^{3}+x^{2}+x}-2 x -1}{x^{2}+1}\right )}{4}\) | \(136\) |
default | \(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}+\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{6 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\) | \(1080\) |
elliptic | \(\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{2 \sqrt {x^{3}+x^{2}+x}}-\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{2 \sqrt {x^{3}+x^{2}+x}}+\frac {3 \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {1}{2}+i+\frac {i \sqrt {3}}{2}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {1}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{12 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{4 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}\) | \(1764\) |
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^{4} - x^{2} + 1}{{\left (x^{4} - 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 = 0.03, size = 698, normalized size = 7.84
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^{4} - x^{2} + 1}{\sqrt {x \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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