Integrand size = 14, antiderivative size = 89 \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\frac {64 a^3 \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
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Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2726, 2725} \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\frac {64 a^3 \sin (c+d x)}{15 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{15 d}+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
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Rule 2725
Rule 2726
Rubi steps \begin{align*} \text {integral}& = \frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} (8 a) \int (a+a \cos (c+d x))^{3/2} \, dx \\ & = \frac {16 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{15} \left (32 a^2\right ) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {64 a^3 \sin (c+d x)}{15 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (150 \sin \left (\frac {1}{2} (c+d x)\right )+25 \sin \left (\frac {3}{2} (c+d x)\right )+3 \sin \left (\frac {5}{2} (c+d x)\right )\right )}{30 d} \]
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Time = 0.88 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {8 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (3 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right ) \sqrt {2}}{15 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(73\) |
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Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\frac {2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{2} + 14 \, a^{2} \cos \left (d x + c\right ) + 43 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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\[ \int (a+a \cos (c+d x))^{5/2} \, dx=\int \left (a \cos {\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \]
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none
Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\frac {{\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 25 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 150 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{30 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94 \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (3 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 25 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 150 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{30 \, d} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \, dx=\int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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