Integrand size = 12, antiderivative size = 70 \[ \int (c \cos (a+b x))^{5/2} \, dx=\frac {6 c^2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}+\frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b} \]
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Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2719} \[ \int (c \cos (a+b x))^{5/2} \, dx=\frac {6 c^2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}+\frac {2 c \sin (a+b x) (c \cos (a+b x))^{3/2}}{5 b} \]
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Rule 2715
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}+\frac {1}{5} \left (3 c^2\right ) \int \sqrt {c \cos (a+b x)} \, dx \\ & = \frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}+\frac {\left (3 c^2 \sqrt {c \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 \sqrt {\cos (a+b x)}} \\ & = \frac {6 c^2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}+\frac {2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89 \[ \int (c \cos (a+b x))^{5/2} \, dx=\frac {(c \cos (a+b x))^{5/2} \left (6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} \sin (2 (a+b x))\right )}{5 b \cos ^{\frac {5}{2}}(a+b x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(212\) vs. \(2(86)=172\).
Time = 2.78 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.04
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c^{3} \left (-8 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, b}\) | \(213\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.30 \[ \int (c \cos (a+b x))^{5/2} \, dx=\frac {2 \, \sqrt {c \cos \left (b x + a\right )} c^{2} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 i \, \sqrt {2} c^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 3 i \, \sqrt {2} c^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{5 \, b} \]
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Timed out. \[ \int (c \cos (a+b x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (c \cos (a+b x))^{5/2} \, dx=\int { \left (c \cos \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (c \cos (a+b x))^{5/2} \, dx=\int { \left (c \cos \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (c \cos (a+b x))^{5/2} \, dx=\int {\left (c\,\cos \left (a+b\,x\right )\right )}^{5/2} \,d x \]
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