Optimal. Leaf size=9 \[ E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3177} \[ E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 3177
Rubi steps
\begin {align*} \int \sqrt {1+\cos ^2(x)} \, dx &=E\left (\left .\frac {\pi }{2}+x\right |-1\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 11, normalized size = 1.22 \[ \sqrt {2} E\left (x\left |\frac {1}{2}\right .\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\cos \relax (x)^{2} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos \relax (x)^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.56, size = 41, normalized size = 4.56 \[ -\frac {\sqrt {\left (1+\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticE \left (\cos \relax (x ), i\right )}{\sqrt {1-\left (\cos ^{4}\relax (x )\right )}\, \sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos \relax (x)^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.01, size = 7, normalized size = 0.78 \[ \sqrt {2}\,\mathrm {E}\left (x\middle |\frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos ^{2}{\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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