Optimal. Leaf size=82 \[ -\frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {3 x+7}{\sqrt {13} \sqrt {x^2+2 x+5}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2074, 724, 206, 1025, 982, 204, 1024} \[ -\frac {\tan ^{-1}\left (\frac {x+1}{\sqrt {3} \sqrt {x^2+2 x+5}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {3 x+7}{\sqrt {13} \sqrt {x^2+2 x+5}}\right )}{12 \sqrt {13}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
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*}
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Mathematica [C] time = 0.36, size = 159, normalized size = 1.94 \[ \frac {1}{312} \left (-2 \sqrt {13} \tanh ^{-1}\left (\frac {3 x+7}{\sqrt {13} \sqrt {x^2+2 x+5}}\right )-13 \left (\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {2 \left (\sqrt [3]{-1}-2\right ) x+5 i \sqrt {3}+1}{\sqrt {2-2 i \sqrt {3}} \sqrt {x^2+2 x+5}}\right )+\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {-2 \left (2+(-1)^{2/3}\right ) x-5 i \sqrt {3}+1}{\sqrt {2+2 i \sqrt {3}} \sqrt {x^2+2 x+5}}\right )\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 119, normalized size = 1.45 \[ \frac {\tan ^{-1}\left (\frac {x^2}{\sqrt {3}}-\frac {(x+1) \sqrt {x^2+2 x+5}}{\sqrt {3}}+\frac {2 x}{\sqrt {3}}+\frac {4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{12} \tanh ^{-1}\left (\sqrt {x^2+2 x+5}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+2 x+5}}{\sqrt {13}}-\frac {x}{\sqrt {13}}+\frac {2}{\sqrt {13}}\right )}{6 \sqrt {13}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 151, normalized size = 1.84 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x + 2\right )} + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} + 2 \, x + 5}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {x^{2} + 2 \, x + 5}\right ) + \frac {1}{156} \, \sqrt {13} \log \left (\frac {\sqrt {13} {\left (3 \, x + 7\right )} + \sqrt {x^{2} + 2 \, x + 5} {\left (3 \, \sqrt {13} - 13\right )} - 9 \, x - 21}{x - 2}\right ) - \frac {1}{24} \, \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 5} {\left (x + 2\right )} + 3 \, x + 6\right ) + \frac {1}{24} \, \log \left (x^{2} - \sqrt {x^{2} + 2 \, x + 5} x + x + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.72, size = 164, normalized size = 2.00 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5} + 2\right )}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}\right ) + \frac {1}{156} \, \sqrt {13} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {13} + 2 \, \sqrt {x^{2} + 2 \, x + 5} + 4 \right |}}{{\left | -2 \, x + 2 \, \sqrt {13} + 2 \, \sqrt {x^{2} + 2 \, x + 5} + 4 \right |}}\right ) - \frac {1}{24} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}^{2} + 4 \, x - 4 \, \sqrt {x^{2} + 2 \, x + 5} + 7\right ) + \frac {1}{24} \, \log \left ({\left (x - \sqrt {x^{2} + 2 \, x + 5}\right )}^{2} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 69, normalized size = 0.84
method | result | size |
default | \(\frac {\arctanh \left (\sqrt {x^{2}+2 x +5}\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +2\right )}{6 \sqrt {x^{2}+2 x +5}}\right )}{12}-\frac {\sqrt {13}\, \arctanh \left (\frac {\left (14+6 x \right ) \sqrt {13}}{26 \sqrt {\left (-2+x \right )^{2}+1+6 x}}\right )}{156}\) | \(69\) |
trager | \(\RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \ln \left (-\frac {-2880 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x +126 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \sqrt {x^{2}+2 x +5}+306 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -8 \sqrt {x^{2}+2 x +5}+570 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )-7 x -19}{12 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -x -2}\right )+\frac {\ln \left (\frac {5760 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x +252 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \sqrt {x^{2}+2 x +5}-348 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -5 \sqrt {x^{2}+2 x +5}+1140 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+3 x -57}{6 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x +1}\right )}{12}-\ln \left (\frac {5760 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x +252 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \sqrt {x^{2}+2 x +5}-348 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -5 \sqrt {x^{2}+2 x +5}+1140 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+3 x -57}{6 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x +1}\right ) \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )-\frac {\RootOf \left (\textit {\_Z}^{2}-13\right ) \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{2}-13\right ) x +13 \sqrt {x^{2}+2 x +5}+7 \RootOf \left (\textit {\_Z}^{2}-13\right )}{-2+x}\right )}{156}\) | \(388\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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