Optimal. Leaf size=86 \[ \frac {14 a \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}-\frac {4 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d} \]
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Rubi [A]
time = 0.08, 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 = {2838, 2830,
2725} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2830
Rule 2838
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 = 0.11, size = 68, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \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 )}{30 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 71, normalized size = 0.83
method | result | size |
default | \(\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (12 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7\right ) \sqrt {2}}{15 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 51, normalized size = 0.59 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 52, normalized size = 0.60 \begin {gather*} \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 )}} \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 \left (\cos {\left (c + d x \right )} + 1\right )} \cos ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 75, normalized size = 0.87 \begin {gather*} \frac {\sqrt {2} {\left (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 ) + 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 ) + 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 )\right )} \sqrt {a}}{30 \, d} \end {gather*}
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 \begin {gather*} \int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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