Integrand size = 21, antiderivative size = 98 \[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {14 b^2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {14 b (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d} \]
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Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2715, 2721, 2719} \[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {14 b^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b d}+\frac {14 b \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d} \]
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Rule 16
Rule 2715
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{9/2} \, dx}{b^2} \\ & = \frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {7}{9} \int (b \cos (c+d x))^{5/2} \, dx \\ & = \frac {14 b (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {1}{15} \left (7 b^2\right ) \int \sqrt {b \cos (c+d x)} \, dx \\ & = \frac {14 b (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d}+\frac {\left (7 b^2 \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}} \\ & = \frac {14 b^2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {14 b (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {(b \cos (c+d x))^{5/2} \left (168 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sqrt {\cos (c+d x)} (38 \sin (2 (c+d x))+5 \sin (4 (c+d x)))\right )}{180 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(110)=220\).
Time = 5.66 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.28
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{3} \left (160 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-480 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+616 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-432 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(223\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {21 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (5 \, b^{2} \cos \left (d x + c\right )^{3} + 7 \, b^{2} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, d} \]
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Timed out. \[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(c+d x) (b \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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