Optimal. Leaf size=62 \[ \frac {2 a (3 A+B) \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 B \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2751, 2646} \[ \frac {2 a (3 A+B) \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 B \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rubi steps
\begin {align*} \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\frac {2 B \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} (3 A+B) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a (3 A+B) \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 B \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 46, normalized size = 0.74 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (3 A+B \cos (c+d x)+2 B)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 47, normalized size = 0.76 \[ \frac {2 \, {\left (B \cos \left (d x + c\right ) + 3 \, A + 2 \, B\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 83, normalized size = 1.34 \[ \frac {1}{3} \, \sqrt {2} {\left (\frac {B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {3 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 62, normalized size = 1.00 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2 B \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 A +B \right ) \sqrt {2}}{3 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 57, normalized size = 0.92 \[ \frac {6 \, \sqrt {2} A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + {\left (\sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a}}{3 \, d} \]
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 \[ \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )} \left (A + B \cos {\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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