Optimal. Leaf size=26 \[ \frac {2 a \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2725}
\begin {gather*} \frac {2 a \sin (c+d x)}{d \sqrt {a \cos (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rubi steps
\begin {align*} \int \sqrt {a+a \cos (c+d x)} \, dx &=\frac {2 a \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.12 \begin {gather*} \frac {2 \sqrt {a (1+\cos (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 43, normalized size = 1.65
method | result | size |
default | \(\frac {2 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}}{\sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(43\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} {\mathrm e}^{-i \left (d x +c \right )}}\, \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{\left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) d}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 20, normalized size = 0.77 \begin {gather*} \frac {2 \, \sqrt {2} \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 32, normalized size = 1.23 \begin {gather*} \frac {2 \, \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d} \end {gather*}
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 \begin {gather*} \int \sqrt {a \cos {\left (c + d x \right )} + a}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 30, normalized size = 1.15 \begin {gather*} \frac {2 \, \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 33, normalized size = 1.27 \begin {gather*} \frac {2\,\sin \left (c+d\,x\right )\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}}{d\,\left (\cos \left (c+d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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