Optimal. Leaf size=14 \[ \frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2670, 14}
\begin {gather*} \frac {\cos ^2(x)}{2}-\log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2670
Rubi steps
\begin {align*} \int \sin ^2(x) \tan (x) \, dx &=-\text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,\cos (x)\right )\\ &=\frac {\cos ^2(x)}{2}-\log (\cos (x))\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {\cos ^2(x)}{2}-\log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 13, normalized size = 0.93
method | result | size |
default | \(-\frac {\left (\sin ^{2}\left (x \right )\right )}{2}-\ln \left (\cos \left (x \right )\right )\) | \(13\) |
risch | \(i x +\frac {{\mathrm e}^{2 i x}}{8}+\frac {{\mathrm e}^{-2 i x}}{8}-\ln \left ({\mathrm e}^{2 i x}+1\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.01, size = 16, normalized size = 1.14 \begin {gather*} -\frac {1}{2} \, \sin \left (x\right )^{2} - \frac {1}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, \cos \left (x\right )^{2} - \log \left (-\cos \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 10, normalized size = 0.71 \begin {gather*} - \log {\left (\cos {\left (x \right )} \right )} + \frac {\cos ^{2}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 18, normalized size = 1.29 \begin {gather*} -\frac {1}{2} \, \sin \left (x\right )^{2} - \frac {1}{2} \, \log \left (-\sin \left (x\right )^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 16, normalized size = 1.14 \begin {gather*} \frac {{\cos \left (x\right )}^2}{2}+\frac {\ln \left ({\mathrm {tan}\left (x\right )}^2+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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