Optimal. Leaf size=99 \[ \frac {1}{3} \sqrt {-3+2 \sqrt {3}} \text {ArcTan}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.59, antiderivative size = 450, normalized size of antiderivative = 4.55, number of steps
used = 14, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1607, 6860,
225, 2160, 2165, 212, 209} \begin {gather*} \frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {\left (2-\sqrt {3}\right )^{3/2} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {1}{3} \sqrt {2 \sqrt {3}-3} \text {ArcTan}\left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {x^3-1}}\right )+\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 212
Rule 225
Rule 1607
Rule 2160
Rule 2165
Rule 6860
Rubi steps
\begin {align*} \int \frac {2 x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx &=\int \frac {x (2+x)}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx\\ &=\int \left (\frac {1}{\sqrt {-1+x^3}}+\frac {2 (1+2 x)}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}}\right ) \, dx\\ &=2 \int \frac {1+2 x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx+\int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+2 \int \left (\frac {2+\sqrt {3}}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}+\frac {2-\sqrt {3}}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}\right ) \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\left (2 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\left (2 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{288} \left (-3+2 \sqrt {3}\right ) \int \frac {96 \left (1+\sqrt {3}\right )-96 x}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\frac {1}{6} \left (-3+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {1}{288} \left (3+2 \sqrt {3}\right ) \int \frac {96 \left (1-\sqrt {3}\right )-96 x}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx-\frac {1}{6} \left (3+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2-\sqrt {3}\right )^{3/2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{3} \left (3-2 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1-\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{-2+2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )+\frac {1}{3} \left (3+2 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1-\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{-2-2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )\\ &=-\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )+\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2-\sqrt {3}\right )^{3/2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.12, size = 99, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt {-3+2 \sqrt {3}} \text {ArcTan}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 2.51, size = 1517, normalized size = 15.32
method | result | size |
trager | \(-\frac {\RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {243 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{5} x^{2}+486 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{5} x -306 x^{2} \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{3}-396 x \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{3}-216 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{3}+48 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \sqrt {x^{3}-1}+95 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right ) x^{2}+38 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right ) x +152 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )-32 \sqrt {x^{3}-1}}{\left (9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} x -5 x -4\right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) \ln \left (\frac {243 \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{4} x^{2}+486 \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{4} x -18 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x^{2}-252 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x +216 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right )-\RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x^{2}-144 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \sqrt {x^{3}-1}-10 \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x +8 \RootOf \left (\textit {\_Z}^{2}+9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right )}{\left (9 \RootOf \left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} x -x +4\right )^{2}}\right )}{6}\) | \(584\) |
default | \(\text {Expression too large to display}\) | \(1517\) |
elliptic | \(\text {Expression too large to display}\) | \(1726\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (75) = 150\).
time = 0.46, size = 218, normalized size = 2.20 \begin {gather*} \frac {1}{3} \, \sqrt {2 \, \sqrt {3} - 3} \arctan \left (\frac {\sqrt {x^{3} - 1} \sqrt {2 \, \sqrt {3} - 3}}{x^{2} + x + 1}\right ) - \frac {1}{12} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x - 1\right )} - 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{12} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x - 1\right )} - 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (x + 2\right )}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - 2 x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.83, size = 509, normalized size = 5.14 \begin {gather*} -\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\left (4\,\sqrt {3}-6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {\left (4\,\sqrt {3}+6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________