Optimal. Leaf size=84 \[ -\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {3} \sqrt {5-2 x+x^2}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {5-2 x+x^2}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {5-2 x+x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2099, 738, 212,
1039, 996, 209, 1038, 213} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2-2 x+5}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {x^2-2 x+5}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^2-2 x+5}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 212
Rule 213
Rule 738
Rule 996
Rule 1038
Rule 1039
Rule 2099
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {5-2 x+x^2} \left (8+x^3\right )} \, dx &=\int \left (\frac {1}{12 (2+x) \sqrt {5-2 x+x^2}}+\frac {4-x}{12 \left (4-2 x+x^2\right ) \sqrt {5-2 x+x^2}}\right ) \, dx\\ &=\frac {1}{12} \int \frac {1}{(2+x) \sqrt {5-2 x+x^2}} \, dx+\frac {1}{12} \int \frac {4-x}{\left (4-2 x+x^2\right ) \sqrt {5-2 x+x^2}} \, dx\\ &=-\left (\frac {1}{24} \int \frac {-2+2 x}{\left (4-2 x+x^2\right ) \sqrt {5-2 x+x^2}} \, dx\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {14-6 x}{\sqrt {5-2 x+x^2}}\right )+\frac {1}{4} \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt {5-2 x+x^2}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {5-2 x+x^2}}\right )}{12 \sqrt {13}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-2+2 x^2} \, dx,x,\sqrt {5-2 x+x^2}\right )+\text {Subst}\left (\int \frac {1}{24+2 x^2} \, dx,x,\frac {-2+2 x}{\sqrt {5-2 x+x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {-2+2 x}{2 \sqrt {3} \sqrt {5-2 x+x^2}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {5-2 x+x^2}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {5-2 x+x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 90, normalized size = 1.07 \begin {gather*} \frac {1}{156} \left (-13 \sqrt {3} \tan ^{-1}\left (\frac {4-2 x+x^2-(-1+x) \sqrt {5-2 x+x^2}}{\sqrt {3}}\right )+13 \tanh ^{-1}\left (\sqrt {5-2 x+x^2}\right )+2 \sqrt {13} \tanh ^{-1}\left (\frac {2+x-\sqrt {5-2 x+x^2}}{\sqrt {13}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.54, size = 69, normalized size = 0.82
method | result | size |
default | \(\frac {\arctanh \left (\sqrt {x^{2}-2 x +5}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2+2 x \right )}{6 \sqrt {x^{2}-2 x +5}}\right )}{12}-\frac {\sqrt {13}\, \arctanh \left (\frac {\left (14-6 x \right ) \sqrt {13}}{26 \sqrt {\left (x +2\right )^{2}-6 x +1}}\right )}{156}\) | \(69\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-13\right ) x -7 \RootOf \left (\textit {\_Z}^{2}-13\right )+13 \sqrt {x^{2}-2 x +5}}{x +2}\right )}{156}-\frac {\ln \left (\frac {5760 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -252 \sqrt {x^{2}-2 x +5}\, \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+348 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +1140 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-5 \sqrt {x^{2}-2 x +5}+3 x +57}{6 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +1}\right )}{12}-\ln \left (\frac {5760 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x -252 \sqrt {x^{2}-2 x +5}\, \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+348 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +1140 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-5 \sqrt {x^{2}-2 x +5}+3 x +57}{6 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +1}\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {2880 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x +126 \sqrt {x^{2}-2 x +5}\, \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+306 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x -570 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+8 \sqrt {x^{2}-2 x +5}+7 x -19}{12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x +x -2}\right )\) | \(384\) |
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. 154 vs.
\(2 (64) = 128\).
time = 0.37, size = 154, normalized size = 1.83 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - 2\right )} + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} - 2 \, x + 5}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} - 2 \, x + 5}\right ) + \frac {1}{156} \, \sqrt {13} \log \left (-\frac {\sqrt {13} {\left (3 \, x - 7\right )} + \sqrt {x^{2} - 2 \, x + 5} {\left (3 \, \sqrt {13} + 13\right )} + 9 \, x - 21}{x + 2}\right ) + \frac {1}{24} \, \log \left (x^{2} - \sqrt {x^{2} - 2 \, x + 5} {\left (x - 2\right )} - 3 \, x + 6\right ) - \frac {1}{24} \, \log \left (x^{2} - \sqrt {x^{2} - 2 \, x + 5} x - 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 {1}{\left (x + 2\right ) \left (x^{2} - 2 x + 4\right ) \sqrt {x^{2} - 2 x + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 164 vs.
\(2 (64) = 128\).
time = 2.14, size = 164, normalized size = 1.95 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} - 2 \, x + 5}\right )}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} - 2 \, x + 5} - 2\right )}\right ) + \frac {1}{156} \, \sqrt {13} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {13} + 2 \, \sqrt {x^{2} - 2 \, x + 5} - 4 \right |}}{{\left | -2 \, x + 2 \, \sqrt {13} + 2 \, \sqrt {x^{2} - 2 \, x + 5} - 4 \right |}}\right ) + \frac {1}{24} \, \log \left ({\left (x - \sqrt {x^{2} - 2 \, x + 5}\right )}^{2} - 4 \, x + 4 \, \sqrt {x^{2} - 2 \, x + 5} + 7\right ) - \frac {1}{24} \, \log \left ({\left (x - \sqrt {x^{2} - 2 \, x + 5}\right )}^{2} + 3\right ) \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^3+8\right )\,\sqrt {x^2-2\,x+5}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________