Optimal. Leaf size=143 \[ -\frac {i \sqrt {-x^4+x^2+2 x-2} \left (-\frac {1}{4} i \tanh ^{-1}\left (-\sqrt {x^2+2 x+2}+x+1\right )-\frac {7 i \tanh ^{-1}\left (\frac {\sqrt {x^2+2 x+2}}{\sqrt {5}}-\frac {x}{\sqrt {5}}+\frac {1}{\sqrt {5}}\right )}{20 \sqrt {5}}-\frac {i \left (x^3-2 x^2-2 x+4\right )}{20 (x-1)^2 \sqrt {x^2+2 x+2}}\right )}{(x-1) \sqrt {x^2+2 x+2}} \]
________________________________________________________________________________________
Rubi [F] time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(1+x) \left (-2+2 x+x^2-x^4\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{(1+x) \left (-2+2 x+x^2-x^4\right )^{3/2}} \, dx &=\int \frac {1}{(1+x) \left (-2+2 x+x^2-x^4\right )^{3/2}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 121, normalized size = 0.85 \begin {gather*} \frac {7 \sqrt {5} \sqrt {x^2+2 x+2} (x-1)^2 \tanh ^{-1}\left (\frac {2 x+3}{\sqrt {5} \sqrt {x^2+2 x+2}}\right )-25 \sqrt {x^2+2 x+2} (x-1)^2 \tanh ^{-1}\left (\sqrt {(x+1)^2+1}\right )+10 \left (x^3-2 x^2-2 x+4\right )}{200 (x-1) \sqrt {-x^4+x^2+2 x-2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 8.74, size = 143, normalized size = 1.00 \begin {gather*} -\frac {i \sqrt {-2+2 x+x^2-x^4} \left (-\frac {i \left (4-2 x-2 x^2+x^3\right )}{20 (-1+x)^2 \sqrt {2+2 x+x^2}}-\frac {1}{4} i \tanh ^{-1}\left (1+x-\sqrt {2+2 x+x^2}\right )-\frac {7 i \tanh ^{-1}\left (\frac {1}{\sqrt {5}}-\frac {x}{\sqrt {5}}+\frac {\sqrt {2+2 x+x^2}}{\sqrt {5}}\right )}{20 \sqrt {5}}\right )}{(-1+x) \sqrt {2+2 x+x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 174, normalized size = 1.22 \begin {gather*} -\frac {7 \, \sqrt {5} {\left (x^{5} - x^{4} - x^{3} - x^{2} + 4 \, x - 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {-x^{4} + x^{2} + 2 \, x - 2} {\left (2 \, x + 3\right )}}{5 \, {\left (x^{3} + x^{2} - 2\right )}}\right ) - 25 \, {\left (x^{5} - x^{4} - x^{3} - x^{2} + 4 \, x - 2\right )} \arctan \left (\frac {\sqrt {-x^{4} + x^{2} + 2 \, x - 2}}{x^{3} + x^{2} - 2}\right ) + 10 \, \sqrt {-x^{4} + x^{2} + 2 \, x - 2} {\left (x^{3} - 2 \, x^{2} - 2 \, x + 4\right )}}{200 \, {\left (x^{5} - x^{4} - x^{3} - x^{2} + 4 \, x - 2\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*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.36, size = 118, normalized size = 0.83
method | result | size |
risch | \(\frac {x^{3}-2 x^{2}-2 x +4}{20 \left (-1+x \right ) \sqrt {-\left (x^{2}+2 x +2\right ) \left (-1+x \right )^{2}}}-\frac {\left (\frac {7 \sqrt {5}\, \arctan \left (\frac {\left (-6-4 x \right ) \sqrt {5}}{10 \sqrt {-\left (-1+x \right )^{2}-1-4 x}}\right )}{200}+\frac {\arctan \left (\frac {1}{\sqrt {-\left (1+x \right )^{2}-1}}\right )}{8}\right ) \left (-1+x \right ) \sqrt {-x^{2}-2 x -2}}{\sqrt {-\left (x^{2}+2 x +2\right ) \left (-1+x \right )^{2}}}\) | \(118\) |
trager | \(-\frac {\left (x^{3}-2 x^{2}-2 x +4\right ) \sqrt {-x^{4}+x^{2}+2 x -2}}{20 \left (-1+x \right )^{3} \left (x^{2}+2 x +2\right )}-\frac {7 \RootOf \left (\textit {\_Z}^{2}+5\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+5\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+5\right ) x +5 \sqrt {-x^{4}+x^{2}+2 x -2}-3 \RootOf \left (\textit {\_Z}^{2}+5\right )}{\left (-1+x \right )^{2}}\right )}{200}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{4}+x^{2}+2 x -2}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right )}\right )}{8}\) | \(160\) |
default | \(\frac {\left (-7 \arctan \left (\frac {\left (3+2 x \right ) \sqrt {5}}{5 \sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, \sqrt {5}\, x^{2}+25 \arctan \left (\frac {1}{\sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, x^{2}+14 \arctan \left (\frac {\left (3+2 x \right ) \sqrt {5}}{5 \sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, \sqrt {5}\, x -50 \arctan \left (\frac {1}{\sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}\, x -7 \sqrt {5}\, \arctan \left (\frac {\left (3+2 x \right ) \sqrt {5}}{5 \sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}-10 x^{3}+25 \arctan \left (\frac {1}{\sqrt {-x^{2}-2 x -2}}\right ) \sqrt {-x^{2}-2 x -2}+20 x^{2}+20 x -40\right ) \left (-1+x \right ) \left (x^{2}+2 x +2\right )}{200 \left (-x^{4}+x^{2}+2 x -2\right )^{\frac {3}{2}}}\) | \(253\) |
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 {1}{{\left (-x^{4} + x^{2} + 2 \, x - 2\right )}^{\frac {3}{2}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x+1\right )\,{\left (-x^4+x^2+2\,x-2\right )}^{3/2}} \,d x \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 {1}{\left (- \left (x - 1\right )^{2} \left (x^{2} + 2 x + 2\right )\right )^{\frac {3}{2}} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________