Optimal. Leaf size=56 \[ \frac {2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}+\frac {2 a \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2751, 2646} \[ \frac {2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}+\frac {2 a \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rubi steps
\begin {align*} \int \cos (c+d x) \sqrt {a+a \cos (c+d x)} \, dx &=\frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 54, normalized size = 0.96 \[ \frac {\left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {3}{2} (c+d x)\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 40, normalized size = 0.71 \[ \frac {2 \, \sqrt {a \cos \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 56, normalized size = 1.00 \[ \frac {1}{3} \, \sqrt {2} \sqrt {a} {\left (\frac {\mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {3 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 58, normalized size = 1.04 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sqrt {2}}{3 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.88, size = 36, normalized size = 0.64 \[ \frac {{\left (\sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \cos \left (c+d\,x\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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