Optimal. Leaf size=84 \[ -\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2-2 x+5}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {x^2-2 x+5}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^2-2 x+5}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, 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 = {2074, 724, 206, 1025, 982, 203, 1024, 207} \[ -\frac {\tan ^{-1}\left (\frac {1-x}{\sqrt {3} \sqrt {x^2-2 x+5}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {x^2-2 x+5}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^2-2 x+5}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 207
Rule 724
Rule 982
Rule 1024
Rule 1025
Rule 2074
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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{-2+2 x^2} \, dx,x,\sqrt {5-2 x+x^2}\right )+\operatorname {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 [C] time = 0.33, size = 160, normalized size = 1.90 \[ \frac {1}{312} \left (-2 \sqrt {13} \tanh ^{-1}\left (\frac {7-3 x}{\sqrt {13} \sqrt {x^2-2 x+5}}\right )-13 \left (\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {-2 \sqrt [3]{-1} x+4 x+5 i \sqrt {3}+1}{\sqrt {2-2 i \sqrt {3}} \sqrt {x^2-2 x+5}}\right )+\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {2 \left (2+(-1)^{2/3}\right ) x-5 i \sqrt {3}+1}{\sqrt {2+2 i \sqrt {3}} \sqrt {x^2-2 x+5}}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.69, size = 154, normalized size = 1.83 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - 2\right )} + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} - 2 \, x + 5}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} - 2 \, x + 5}\right ) + \frac {1}{156} \, \sqrt {13} \log \left (-\frac {\sqrt {13} {\left (3 \, x - 7\right )} + \sqrt {x^{2} - 2 \, x + 5} {\left (3 \, \sqrt {13} + 13\right )} + 9 \, x - 21}{x + 2}\right ) + \frac {1}{24} \, \log \left (x^{2} - \sqrt {x^{2} - 2 \, x + 5} {\left (x - 2\right )} - 3 \, x + 6\right ) - \frac {1}{24} \, \log \left (x^{2} - \sqrt {x^{2} - 2 \, x + 5} x - x + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.59, size = 164, normalized size = 1.95 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} - 2 \, x + 5}\right )}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} - 2 \, x + 5} - 2\right )}\right ) + \frac {1}{156} \, \sqrt {13} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {13} + 2 \, \sqrt {x^{2} - 2 \, x + 5} - 4 \right |}}{{\left | -2 \, x + 2 \, \sqrt {13} + 2 \, \sqrt {x^{2} - 2 \, x + 5} - 4 \right |}}\right ) + \frac {1}{24} \, \log \left ({\left (x - \sqrt {x^{2} - 2 \, x + 5}\right )}^{2} - 4 \, x + 4 \, \sqrt {x^{2} - 2 \, x + 5} + 7\right ) - \frac {1}{24} \, \log \left ({\left (x - \sqrt {x^{2} - 2 \, x + 5}\right )}^{2} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 69, normalized size = 0.82 \[ \frac {\arctanh \left (\sqrt {x^{2}-2 x +5}\right )}{12}-\frac {\sqrt {13}\, \arctanh \left (\frac {\left (-6 x +14\right ) \sqrt {13}}{26 \sqrt {-6 x +\left (x +2\right )^{2}+1}}\right )}{156}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x -2\right )}{6 \sqrt {x^{2}-2 x +5}}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 8\right )} \sqrt {x^{2} - 2 \, x + 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (x^3+8\right )\,\sqrt {x^2-2\,x+5}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + 2\right ) \left (x^{2} - 2 x + 4\right ) \sqrt {x^{2} - 2 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________