Optimal. Leaf size=16 \[ \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 16, 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 = {2639} \[ \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2639
Rubi steps
\begin {align*} \int \sqrt {\cos (a+b x)} \, dx &=\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\cos \left (b x + a\right )}, x\right ) \]
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 \[ \int \sqrt {\cos \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 133, normalized size = 8.31 \[ \frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]
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 \[ \int \sqrt {\cos \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 15, normalized size = 0.94 \[ \frac {2\,\mathrm {E}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{b} \]
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 \[ \int \sqrt {\cos {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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