Optimal. Leaf size=89 \[ \frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {1}{4} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right ) \]
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Rubi [A] time = 3.13, antiderivative size = 155, normalized size of antiderivative = 1.74, number of steps used = 41, number of rules used = 18, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2056, 6725, 918, 27, 6733, 1712, 1197, 1103, 1195, 1700, 1698, 206, 12, 1210, 203, 1714, 1708, 1706} \begin {gather*} \frac {\sqrt {x^3+x^2+x} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{4 \sqrt {x} \sqrt {x^2+x+1}}+\frac {\sqrt {x^3+x^2+x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2+x+1}}\right )}{2 \sqrt {x} \sqrt {x^2+x+1}}-\frac {\sqrt {3} \sqrt {x^3+x^2+x} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {x^2+x+1}}\right )}{4 \sqrt {x} \sqrt {x^2+x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 203
Rule 206
Rule 918
Rule 1103
Rule 1195
Rule 1197
Rule 1210
Rule 1698
Rule 1700
Rule 1706
Rule 1708
Rule 1712
Rule 1714
Rule 2056
Rule 6725
Rule 6733
Rubi steps
\begin {align*} \int \frac {\sqrt {x+x^2+x^3}}{-1+x^4} \, dx &=\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{-1+x^4} \, dx}{\sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\sqrt {x+x^2+x^3} \int \left (-\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 \left (1-x^2\right )}-\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 \left (1+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1-x^2} \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1+x^2} \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {\sqrt {x+x^2+x^3} \int \left (\frac {i \sqrt {x} \sqrt {1+x+x^2}}{2 (i-x)}+\frac {i \sqrt {x} \sqrt {1+x+x^2}}{2 (i+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \left (\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 (1-x)}+\frac {\sqrt {x} \sqrt {1+x+x^2}}{2 (1+x)}\right ) \, dx}{2 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{i-x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{i+x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1-x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {\sqrt {x} \sqrt {1+x+x^2}}{1+x} \, dx}{4 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {i-(2-2 i) x-(1-3 i) x^2}{\sqrt {x} (i+x) \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \int \frac {i+(2+2 i) x+(1+3 i) x^2}{(i-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \int \frac {1+2 x^2}{\sqrt {x} (1+x) \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {1+4 x+4 x^2}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\left (i \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {i-(2-2 i) x^2-(1-3 i) x^4}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (i \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {i+(2+2 i) x^2+(1+3 i) x^4}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \int \frac {(1+2 x)^2}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1+2 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\left (-\frac {\left (\left (\frac {1}{2}-\frac {i}{6}\right ) \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}\right )+\frac {\left (i \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-3-6 i x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (i \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {-3+6 i x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {3}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {\left (1+2 x^2\right )^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{6}\right ) \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {2 \sqrt {x+x^2+x^3}}{3 (1+x)}+\frac {2 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{3 \sqrt {x} \left (1+x+x^2\right )}+-\frac {\left (\left (\frac {3}{4}+\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+-\frac {\left (\left (\frac {3}{4}-\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+-\frac {\left (\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {5+4 x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (2 \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}\\ &=-\frac {2 \sqrt {x+x^2+x^3}}{3 (1+x)}+\frac {\sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {2 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} E\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{3 \sqrt {x} \left (1+x+x^2\right )}-\frac {3 (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \sqrt {x+x^2+x^3} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{4 \sqrt {x} \left (1+x+x^2\right )}-\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{12 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (2 \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (3 \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\left (4 \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\left (3 \sqrt {x+x^2+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}\\ &=\frac {\sqrt {x+x^2+x^3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}+\frac {\sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )}{2 \sqrt {x} \sqrt {1+x+x^2}}-\frac {\sqrt {3} \sqrt {x+x^2+x^3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )}{4 \sqrt {x} \sqrt {1+x+x^2}}\\ \end {align*}
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Mathematica [C] time = 3.22, size = 743, normalized size = 8.35 \begin {gather*} \frac {(-1)^{2/3} \left (\sqrt {x}-1\right ) \left (\sqrt {x}+1\right ) \sqrt {x} (x+1) \left (x^2+1\right ) \left (\sqrt {\sqrt [3]{-1} x+1} \sqrt {1-(-1)^{2/3} x} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left ((-1)^{5/6} \sqrt {x}\right )|(-1)^{2/3}\right )+\sqrt {\sqrt [3]{-1} x+1} \sqrt {1-(-1)^{2/3} x} \Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} \sqrt {x}\right )|(-1)^{2/3}\right )+\sqrt {\sqrt [3]{-1} x+1} \sqrt {1-(-1)^{2/3} x} \Pi \left ((-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} \sqrt {x}\right )|(-1)^{2/3}\right )+\frac {3 \sqrt {\frac {(-1)^{2/3}-\sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}} \sqrt {\frac {1-\left (\sqrt [3]{-1}-1\right ) \sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}} \sqrt {-\frac {\sqrt {x}+(-1)^{2/3}}{\left (\sqrt [3]{-1}-1\right ) \sqrt {x}+1}} \left (\sqrt [3]{-1}-\sqrt {x}\right )^2 \left (\left (1+\sqrt [3]{-1}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {1-\left (-1+\sqrt [3]{-1}\right ) \sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (-1;\left .\sin ^{-1}\left (\sqrt {\frac {1-\left (-1+\sqrt [3]{-1}\right ) \sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}}\right )\right |-3\right )\right )}{\left (\sqrt [3]{-1}-1\right )^2}-\frac {3 \sqrt {\frac {(-1)^{2/3}-\sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}} \sqrt {\frac {1-\left (\sqrt [3]{-1}-1\right ) \sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}} \sqrt {-\frac {\sqrt {x}+(-1)^{2/3}}{\left (\sqrt [3]{-1}-1\right ) \sqrt {x}+1}} \left (\sqrt [3]{-1}-\sqrt {x}\right )^2 \left (\left (\sqrt [3]{-1}-1\right ) F\left (\left .\sin ^{-1}\left (\sqrt {\frac {1-\left (-1+\sqrt [3]{-1}\right ) \sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}}\right )\right |-3\right )-2 \sqrt [3]{-1} \Pi \left (3;\left .\sin ^{-1}\left (\sqrt {\frac {1-\left (-1+\sqrt [3]{-1}\right ) \sqrt {x}}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-\sqrt {x}\right )}}\right )\right |-3\right )\right )}{\left (\sqrt [3]{-1}-1\right )^2}\right )}{2 \sqrt {x \left (x^2+x+1\right )} \left (x^4-1\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.13, size = 89, normalized size = 1.00 \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right )-\frac {1}{4} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {x^3+x^2+x}}{x^2+x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 120, normalized size = 1.35 \begin {gather*} \frac {1}{16} \, \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 {1}{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 {\sqrt {x^{3} + x^{2} + x}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 1788, normalized size = 20.09
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 {\sqrt {x^{3} + x^{2} + x}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 565, normalized size = 6.35
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 {\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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