Integrand size = 23, antiderivative size = 146 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {832 a^3 \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {208 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d} \]
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Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2838, 2830, 2726, 2725} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {832 a^3 \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {208 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac {4 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac {26 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d} \]
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Rule 2725
Rule 2726
Rule 2830
Rule 2838
Rubi steps \begin{align*} \text {integral}& = \frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {2 \int \left (\frac {7 a}{2}-a \cos (c+d x)\right ) (a+a \cos (c+d x))^{5/2} \, dx}{9 a} \\ & = -\frac {4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {13}{21} \int (a+a \cos (c+d x))^{5/2} \, dx \\ & = \frac {26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {1}{105} (104 a) \int (a+a \cos (c+d x))^{3/2} \, dx \\ & = \frac {208 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {1}{315} \left (416 a^2\right ) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {832 a^3 \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {208 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {26 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}-\frac {4 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (8190 \sin \left (\frac {1}{2} (c+d x)\right )+2100 \sin \left (\frac {3}{2} (c+d x)\right )+756 \sin \left (\frac {5}{2} (c+d x)\right )+225 \sin \left (\frac {7}{2} (c+d x)\right )+35 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{2520 d} \]
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Time = 2.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (140 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+39 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+52 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+104\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(99\) |
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Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.60 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {2 \, {\left (35 \, a^{2} \cos \left (d x + c\right )^{4} + 130 \, a^{2} \cos \left (d x + c\right )^{3} + 219 \, a^{2} \cos \left (d x + c\right )^{2} + 292 \, a^{2} \cos \left (d x + c\right ) + 584 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Time = 0.47 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {{\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]
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Time = 0.81 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (35 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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