Optimal. Leaf size=17 \[ -\frac {x}{2}+\frac {1}{4} \tanh ^{-1}(2 \cos (x) \sin (x)) \]
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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {304, 209, 212}
\begin {gather*} \frac {1}{4} \tanh ^{-1}(2 \sin (x) \cos (x))-\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rubi steps
\begin {align*} \int \sec (2 x) \sin ^2(x) \, dx &=\text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tan (x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=-\frac {x}{2}+\frac {1}{4} \tanh ^{-1}(2 \cos (x) \sin (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.65 \begin {gather*} -\frac {x}{2}-\frac {1}{4} \log (\cos (x)-\sin (x))+\frac {1}{4} \log (\cos (x)+\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 21, normalized size = 1.24
method | result | size |
default | \(-\frac {\arctan \left (\tan \left (x \right )\right )}{2}+\frac {\ln \left (\tan \left (x \right )+1\right )}{4}-\frac {\ln \left (-1+\tan \left (x \right )\right )}{4}\) | \(21\) |
risch | \(-\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{2 i x}-i\right )}{4}+\frac {\ln \left ({\mathrm e}^{2 i x}+i\right )}{4}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (13) = 26\).
time = 1.47, size = 128, normalized size = 7.53 \begin {gather*} -\frac {1}{2} \, x - \frac {1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) + \frac {1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) + \frac {1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{8} \, \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 26, normalized size = 1.53 \begin {gather*} -\frac {1}{2} \, x + \frac {1}{8} \, \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - \frac {1}{8} \, \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 22, normalized size = 1.29 \begin {gather*} - \frac {x}{2} - \frac {\log {\left (\sin {\left (2 x \right )} - 1 \right )}}{8} + \frac {\log {\left (\sin {\left (2 x \right )} + 1 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 20, normalized size = 1.18 \begin {gather*} -\frac {1}{2} \, x + \frac {1}{4} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | \tan \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 = 0.24, size = 9, normalized size = 0.53 \begin {gather*} \frac {\mathrm {atanh}\left (\mathrm {tan}\left (x\right )\right )}{2}-\frac {x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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