Optimal. Leaf size=59 \[ -\frac {\tan ^{-1}\left (\frac {7-40 x}{5 \sqrt {11}}\right )}{2 \sqrt {11}}+\frac {\tan ^{-1}\left (\frac {57+30 x-40 x^2+800 x^3}{6 \sqrt {11}}\right )}{2 \sqrt {11}} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2115}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {800 x^3-40 x^2+30 x+57}{6 \sqrt {11}}\right )}{2 \sqrt {11}}-\frac {\text {ArcTan}\left (\frac {7-40 x}{5 \sqrt {11}}\right )}{2 \sqrt {11}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2115
Rubi steps
\begin {align*} \int \frac {3+12 x+20 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx &=-\frac {\tan ^{-1}\left (\frac {7-40 x}{5 \sqrt {11}}\right )}{2 \sqrt {11}}+\frac {\tan ^{-1}\left (\frac {57+30 x-40 x^2+800 x^3}{6 \sqrt {11}}\right )}{2 \sqrt {11}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 86, normalized size = 1.46 \begin {gather*} \frac {1}{8} \text {RootSum}\left [9+24 \text {$\#$1}-12 \text {$\#$1}^2+80 \text {$\#$1}^3+320 \text {$\#$1}^4\&,\frac {3 \log (x-\text {$\#$1})+12 \log (x-\text {$\#$1}) \text {$\#$1}+20 \log (x-\text {$\#$1}) \text {$\#$1}^2}{3-3 \text {$\#$1}+30 \text {$\#$1}^2+160 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.03, size = 62, normalized size = 1.05
method | result | size |
risch | \(\frac {\sqrt {11}\, \arctan \left (\frac {\left (40 x -7\right ) \sqrt {11}}{55}\right )}{22}+\frac {\sqrt {11}\, \arctan \left (-\frac {20 \sqrt {11}\, x^{2}}{33}+\frac {5 \sqrt {11}\, x}{11}+\frac {19 \sqrt {11}}{22}+\frac {400 \sqrt {11}\, x^{3}}{33}\right )}{22}\) | \(52\) |
default | \(\frac {i \sqrt {11}\, \ln \left (80 x^{2}+\left (10 i \sqrt {11}+10\right ) x +3 i \sqrt {11}-9\right )}{44}-\frac {i \sqrt {11}\, \ln \left (80 x^{2}+\left (-10 i \sqrt {11}+10\right ) x -3 i \sqrt {11}-9\right )}{44}\) | \(62\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 43, normalized size = 0.73 \begin {gather*} \frac {1}{22} \, \sqrt {11} \arctan \left (\frac {1}{66} \, \sqrt {11} {\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) + \frac {1}{22} \, \sqrt {11} \arctan \left (\frac {1}{55} \, \sqrt {11} {\left (40 \, x - 7\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 73, normalized size = 1.24 \begin {gather*} \frac {\sqrt {11} \cdot \left (2 \operatorname {atan}{\left (\frac {8 \sqrt {11} x}{11} - \frac {7 \sqrt {11}}{55} \right )} + 2 \operatorname {atan}{\left (\frac {400 \sqrt {11} x^{3}}{33} - \frac {20 \sqrt {11} x^{2}}{33} + \frac {5 \sqrt {11} x}{11} + \frac {19 \sqrt {11}}{22} \right )}\right )}{44} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.29, size = 40, normalized size = 0.68 \begin {gather*} \frac {1}{22} \, \sqrt {11} {\left (\arctan \left (\frac {1}{66} \, \sqrt {11} {\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) - \arctan \left (-\frac {1}{55} \, \sqrt {11} {\left (40 \, x - 7\right )}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 53, normalized size = 0.90 \begin {gather*} \frac {\sqrt {11}\,\mathrm {atan}\left (\frac {8\,\sqrt {11}\,x}{11}-\frac {7\,\sqrt {11}}{55}\right )}{22}+\frac {\sqrt {11}\,\mathrm {atan}\left (\frac {400\,\sqrt {11}\,x^3}{33}-\frac {20\,\sqrt {11}\,x^2}{33}+\frac {5\,\sqrt {11}\,x}{11}+\frac {19\,\sqrt {11}}{22}\right )}{22} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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