Optimal. Leaf size=86 \[ \frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac {4 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{15 d}+\frac {14 a \sin (c+d x)}{15 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.11, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2759, 2751, 2646} \[ \frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac {4 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{15 d}+\frac {14 a \sin (c+d x)}{15 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx &=\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {2 \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{5 a}\\ &=-\frac {4 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {7}{15} \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {14 a \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}-\frac {4 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 68, normalized size = 0.79 \[ \frac {\left (30 \sin \left (\frac {1}{2} (c+d x)\right )+5 \sin \left (\frac {3}{2} (c+d x)\right )+3 \sin \left (\frac {5}{2} (c+d x)\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{30 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 52, normalized size = 0.60 \[ \frac {2 \, \sqrt {a \cos \left (d x + c\right ) + a} {\left (3 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 81, normalized size = 0.94 \[ \frac {1}{30} \, \sqrt {2} \sqrt {a} {\left (\frac {3 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {5 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {30 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 71, normalized size = 0.83 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (12 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7\right ) \sqrt {2}}{15 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 51, normalized size = 0.59 \[ \frac {{\left (3 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 30 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{30 \, d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )} \cos ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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