Optimal. Leaf size=22 \[ -\frac{1-\sqrt{2} \sin (z)}{\cos (z)-\sin (z)} \]
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Rubi [A] time = 0.135323, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3114} \[ -\frac{1-\sqrt{2} \sin (z)}{\cos (z)-\sin (z)} \]
Antiderivative was successfully verified.
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Rule 3114
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2}+\cos (z)+\sin (z)} \, dz &=-\frac{1-\sqrt{2} \sin (z)}{\cos (z)-\sin (z)}\\ \end{align*}
Mathematica [C] time = 0.0607091, size = 77, normalized size = 3.5 \[ \frac{\left ((1+i)-i \sqrt{2}\right ) \sin \left (\frac{z}{2}\right )-\left (\sqrt{2}+(1+3 i)\right ) \cos \left (\frac{z}{2}\right )}{i \left (\sqrt{2}+(-1-i)\right ) \sin \left (\frac{z}{2}\right )+\left (\sqrt{2}+(1+i)\right ) \cos \left (\frac{z}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 21, normalized size = 1. \begin{align*} -2\,{\frac{1}{ \left ( \sqrt{2}-1 \right ) \left ( \tan \left ( z/2 \right ) +\sqrt{2}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43676, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\frac{{\left (\sqrt{2} - 1\right )} \sin \left (z\right )}{\cos \left (z\right ) + 1} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02825, size = 82, normalized size = 3.73 \begin{align*} \frac{\sqrt{2} \cos \left (z\right ) + \sqrt{2} \sin \left (z\right ) - 2}{2 \,{\left (\cos \left (z\right ) - \sin \left (z\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.84537, size = 51, normalized size = 2.32 \begin{align*} - \frac{2 \tan{\left (\frac{z}{2} \right )}}{- \tan{\left (\frac{z}{2} \right )} + \sqrt{2} \tan{\left (\frac{z}{2} \right )} + 1} + \frac{2 \sqrt{2} \tan{\left (\frac{z}{2} \right )}}{- \tan{\left (\frac{z}{2} \right )} + \sqrt{2} \tan{\left (\frac{z}{2} \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1178, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (\sqrt{2} + 1\right )}}{\sqrt{2} + \tan \left (\frac{1}{2} \, z\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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