Optimal. Leaf size=5 \[ \log (\sin (x)+1) \]
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Rubi [A] time = 0.0244933, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3159, 2667, 31} \[ \log (\sin (x)+1) \]
Antiderivative was successfully verified.
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Rule 3159
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\sec (x)+\tan (x)} \, dx &=\int \frac{\cos (x)}{1+\sin (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sin (x)\right )\\ &=\log (1+\sin (x))\\ \end{align*}
Mathematica [B] time = 0.0138217, size = 16, normalized size = 3.2 \[ 2 \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( 1+\sin \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.979022, size = 42, normalized size = 8.4 \begin{align*} 2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1763, size = 23, normalized size = 4.6 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.165543, size = 17, normalized size = 3.4 \begin{align*} \log{\left (\tan{\left (x \right )} + \sec{\left (x \right )} \right )} - \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12336, size = 30, normalized size = 6. \begin{align*} -\log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + 2 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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