Optimal. Leaf size=22 \[ -\frac {1-\sqrt {2} \sin (z)}{\cos (z)-\sin (z)} \]
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Rubi [A] time = 0.14, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.07, size = 77, normalized size = 3.50 \[ \frac {\left ((1+i)-i \sqrt {2}\right ) \sin \left (\frac {z}{2}\right )-\left (\sqrt {2}+(1+3 i)\right ) \cos \left (\frac {z}{2}\right )}{i \left (\sqrt {2}+(-1-i)\right ) \sin \left (\frac {z}{2}\right )+\left (\sqrt {2}+(1+i)\right ) \cos \left (\frac {z}{2}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 25, normalized size = 1.14 \[ \frac {\sqrt {2} \cos \relax (z) + \sqrt {2} \sin \relax (z) - 2}{2 \, {\left (\cos \relax (z) - \sin \relax (z)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 18, normalized size = 0.82 \[ -\frac {2 \, {\left (\sqrt {2} + 1\right )}}{\sqrt {2} + \tan \left (\frac {1}{2} \, z\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 21, normalized size = 0.95 \[ -\frac {2}{\left (\sqrt {2}-1\right ) \left (\tan \left (\frac {z}{2}\right )+\sqrt {2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 20, normalized size = 0.91 \[ -\frac {2}{\frac {{\left (\sqrt {2} - 1\right )} \sin \relax (z)}{\cos \relax (z) + 1} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 16, normalized size = 0.73 \[ -\frac {2}{\mathrm {tan}\left (\frac {z}{2}\right )\,\left (\sqrt {2}-1\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.73, size = 61, normalized size = 2.77 \[ - \frac {198}{- 239 \tan {\left (\frac {z}{2} \right )} + 169 \sqrt {2} \tan {\left (\frac {z}{2} \right )} - 70 \sqrt {2} + 99} + \frac {140 \sqrt {2}}{- 239 \tan {\left (\frac {z}{2} \right )} + 169 \sqrt {2} \tan {\left (\frac {z}{2} \right )} - 70 \sqrt {2} + 99} \]
Verification of antiderivative is not currently implemented for this CAS.
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