Integrand size = 23, antiderivative size = 159 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {4096 a^5 \cos ^3(c+d x)}{3465 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt {a+a \sin (c+d x)}}-\frac {128 a^3 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{231 d}-\frac {32 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{99 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 d} \]
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Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752} \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {4096 a^5 \cos ^3(c+d x)}{3465 d (a \sin (c+d x)+a)^{3/2}}-\frac {1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt {a \sin (c+d x)+a}}-\frac {128 a^3 \cos ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{231 d}-\frac {32 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{99 d}-\frac {2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 d} \]
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Rule 2752
Rule 2753
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 d}+\frac {1}{11} (16 a) \int \cos ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx \\ & = -\frac {32 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{99 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 d}+\frac {1}{33} \left (64 a^2\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \\ & = -\frac {128 a^3 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{231 d}-\frac {32 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{99 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 d}+\frac {1}{231} \left (512 a^3\right ) \int \cos ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt {a+a \sin (c+d x)}}-\frac {128 a^3 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{231 d}-\frac {32 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{99 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 d}+\frac {\left (2048 a^4\right ) \int \frac {\cos ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{1155} \\ & = -\frac {4096 a^5 \cos ^3(c+d x)}{3465 d (a+a \sin (c+d x))^{3/2}}-\frac {1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt {a+a \sin (c+d x)}}-\frac {128 a^3 \cos ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{231 d}-\frac {32 a^2 \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{99 d}-\frac {2 a \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{11 d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.52 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {2 a^3 \cos ^3(c+d x) \sqrt {a (1+\sin (c+d x))} \left (5419+6396 \sin (c+d x)+4530 \sin ^2(c+d x)+1820 \sin ^3(c+d x)+315 \sin ^4(c+d x)\right )}{3465 d (1+\sin (c+d x))^2} \]
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Time = 1.80 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{4} \left (\sin \left (d x +c \right )-1\right )^{2} \left (315 \left (\sin ^{4}\left (d x +c \right )\right )+1820 \left (\sin ^{3}\left (d x +c \right )\right )+4530 \left (\sin ^{2}\left (d x +c \right )\right )+6396 \sin \left (d x +c \right )+5419\right )}{3465 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(87\) |
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Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.21 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (315 \, a^{3} \cos \left (d x + c\right )^{6} + 1505 \, a^{3} \cos \left (d x + c\right )^{5} - 2150 \, a^{3} \cos \left (d x + c\right )^{4} - 4876 \, a^{3} \cos \left (d x + c\right )^{3} + 512 \, a^{3} \cos \left (d x + c\right )^{2} - 2048 \, a^{3} \cos \left (d x + c\right ) - 4096 \, a^{3} + {\left (315 \, a^{3} \cos \left (d x + c\right )^{5} - 1190 \, a^{3} \cos \left (d x + c\right )^{4} - 3340 \, a^{3} \cos \left (d x + c\right )^{3} + 1536 \, a^{3} \cos \left (d x + c\right )^{2} + 2048 \, a^{3} \cos \left (d x + c\right ) + 4096 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3465 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {64 \, \sqrt {2} {\left (315 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1540 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2970 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2772 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1155 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )} \sqrt {a}}{3465 \, d} \]
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Timed out. \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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