Optimal. Leaf size=33 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {x^3+1}}{\sqrt {2} \left (x^2-x+1\right )}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2145, 203} \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {x+1}{\sqrt {2} \sqrt {x^3+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 2145
Rubi steps
\begin {align*} \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{4+2 x^2} \, dx,x,\frac {1+x}{\sqrt {1+x^3}}\right )\right )\\ &=-\sqrt {2} \tan ^{-1}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^3}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.87, size = 515, normalized size = 15.61 \begin {gather*} \frac {\sqrt {\frac {x+1}{1+\sqrt [3]{-1}}} \left (\frac {2 i \sqrt {23} \sqrt {x^2-x+1} \Pi \left (-\frac {4 \sqrt {3}}{i-2 \sqrt {3}+\sqrt {23}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{i-2 \sqrt {3}+\sqrt {23}}+\frac {10 \sqrt {x^2-x+1} \Pi \left (-\frac {4 \sqrt {3}}{i-2 \sqrt {3}+\sqrt {23}};\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) x-2 i}{-3 i+\sqrt {3}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{i-2 \sqrt {3}+\sqrt {23}}+\frac {10 (-1)^{5/6} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2-x+1} \Pi \left (\frac {4 \sqrt {3}}{-i+2 \sqrt {3}+\sqrt {23}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{6-i \sqrt {3}+\sqrt {69}}+\frac {2 i \sqrt {23} \sqrt {x^2-x+1} \Pi \left (\frac {4 \sqrt {3}}{-i+2 \sqrt {3}+\sqrt {23}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{-i+2 \sqrt {3}+\sqrt {23}}-\frac {2 \left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}\right )}{2 \sqrt {x^3+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.09, size = 33, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {1+x^3}}{\sqrt {2} \left (1-x+x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 28, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - 3 \, x + 1\right )}}{4 \, \sqrt {x^{3} + 1}}\right ) \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*} \int \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (2 \, x^{2} - x + 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.56, size = 62, normalized size = 1.88
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{3}+1}-\RootOf \left (\textit {\_Z}^{2}+2\right )}{2 x^{2}-x +3}\right )}{2}\) | \(62\) |
default | \(\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {\left (\frac {5}{4}+\frac {i \sqrt {23}}{4}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}+\frac {i \sqrt {23}}{12}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}-\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {\left (\frac {5}{4}-\frac {i \sqrt {23}}{4}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}-\frac {i \sqrt {23}}{12}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}+\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(432\) |
elliptic | \(\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{8}+\frac {i \sqrt {23}}{8}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}+\frac {i \sqrt {23}}{12}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}-\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{8}-\frac {i \sqrt {23}}{8}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {5}{12}-\frac {i \sqrt {23}}{12}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}+\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(434\) |
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*} \int \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (2 \, x^{2} - x + 3\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 274, normalized size = 8.30 \begin {gather*} -\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\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+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\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 )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {5}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}};\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 )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {5}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}};\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 )\right )}{2\,\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]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2 x - 2}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x^{2} - x + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________