Optimal. Leaf size=9 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {212}
\begin {gather*} \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rubi steps
\begin {align*} \int \cos ^2(x) \sec (3 x) \, dx &=\text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \tanh ^{-1}(2 \sin (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 9, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(19\) vs.
\(2(7)=14\).
time = 0.08, size = 20, normalized size = 2.22
method | result | size |
default | \(\frac {\ln \left (1+2 \sin \left (x \right )\right )}{4}-\frac {\ln \left (2 \sin \left (x \right )-1\right )}{4}\) | \(20\) |
risch | \(-\frac {\ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}+\frac {\ln \left (i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}\) | \(38\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs.
\(2 (7) = 14\).
time = 1.08, size = 19, normalized size = 2.11 \begin {gather*} \frac {1}{4} \, \log \left (2 \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (-2 \, \sin \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (7) = 14\).
time = 2.09, size = 76, normalized size = 8.44 \begin {gather*} - \frac {\log {\left (\sin {\left (3 x \right )} - 1 \right )}}{12} + \frac {\log {\left (\sin {\left (3 x \right )} + 1 \right )}}{12} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{6} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{6} - \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} - 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (7) = 14\).
time = 0.62, size = 21, normalized size = 2.33 \begin {gather*} \frac {1}{4} \, \log \left ({\left | 2 \, \sin \left (x\right ) + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | 2 \, \sin \left (x\right ) - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 7, normalized size = 0.78 \begin {gather*} \frac {\mathrm {atanh}\left (2\,\sin \left (x\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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