Optimal. Leaf size=95 \[ \frac {2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac {4 C \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{15 d} \]
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Rubi [A] time = 0.13, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3024, 2751, 2646} \[ \frac {2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac {4 C \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{15 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 3024
Rubi steps
\begin {align*} \int \sqrt {a+a \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {2 \int \sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (5 A+3 C)-a C \cos (c+d x)\right ) \, dx}{5 a}\\ &=-\frac {4 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {1}{15} (15 A+7 C) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}-\frac {4 C \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 58, normalized size = 0.61 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (30 A+8 C \cos (c+d x)+3 C \cos (2 (c+d x))+19 C)}{15 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 59, normalized size = 0.62 \[ \frac {2 \, {\left (3 \, C \cos \left (d x + c\right )^{2} + 4 \, C \cos \left (d x + c\right ) + 15 \, A + 8 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 99, normalized size = 1.04 \[ \frac {1}{30} \, \sqrt {2} {\left (\frac {3 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {30 \, {\left (2 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 78, normalized size = 0.82 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (12 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 A +7 C \right ) \sqrt {2}}{15 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 72, normalized size = 0.76 \[ \frac {60 \, \sqrt {2} A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + {\left (3 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 30 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )} \left (A + C \cos ^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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