Optimal. Leaf size=24 \[ \frac {x}{6}-\frac {1}{6} \arctan \left (\frac {\cos (x) \sin (x)}{2+\cos ^2(x)}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {209}
\begin {gather*} \frac {x}{6}-\frac {1}{6} \arctan \left (\frac {\sin (x) \cos (x)}{\cos ^2(x)+2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {1}{9+4 x^2} \, dx,x,\tan (x)\right )\\ &=\frac {x}{6}-\frac {1}{6} \arctan \left (\frac {\cos (x) \sin (x)}{2+\cos ^2(x)}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 0.46 \begin {gather*} \frac {1}{6} \arctan \left (\frac {2 \tan (x)}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 8, normalized size = 0.33
method | result | size |
default | \(\frac {\arctan \left (\frac {2 \tan \left (x \right )}{3}\right )}{6}\) | \(8\) |
risch | \(-\frac {i \ln \left ({\mathrm e}^{2 i x}+\frac {1}{5}\right )}{12}+\frac {i \ln \left ({\mathrm e}^{2 i x}+5\right )}{12}\) | \(24\) |
parallelrisch | \(-\frac {i \left (\ln \left (\frac {-2 i \sin \left (x \right )-3 \cos \left (x \right )}{1+\cos \left (x \right )}\right )-\ln \left (\frac {2 i \sin \left (x \right )-3 \cos \left (x \right )}{1+\cos \left (x \right )}\right )\right )}{12}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 7, normalized size = 0.29 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {2}{3} \, \tan \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 21, normalized size = 0.88 \begin {gather*} -\frac {1}{12} \, \arctan \left (\frac {13 \, \cos \left (x\right )^{2} - 4}{12 \, \cos \left (x\right ) \sin \left (x\right )}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{4 \sin ^{2}{\left (x \right )} + 9 \cos ^{2}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 20, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, x - \frac {1}{6} \, \arctan \left (\frac {\sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 16, normalized size = 0.67 \begin {gather*} \frac {x}{6}-\frac {\mathrm {atan}\left (\mathrm {tan}\left (x\right )\right )}{6}+\frac {\mathrm {atan}\left (\frac {2\,\mathrm {tan}\left (x\right )}{3}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 10, normalized size = 0.42 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {5}\, \sin \left (x \right )}{3}\right )}{3} \end {gather*}
Warning: Unable to verify antiderivative.
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