Integrand size = 19, antiderivative size = 97 \[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {10 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
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Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {16, 2715, 2721, 2720} \[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {10 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 d}+\frac {2 \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d} \]
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Rule 16
Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{7/2} \, dx}{b} \\ & = \frac {2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} (5 b) \int (b \cos (c+d x))^{3/2} \, dx \\ & = \frac {10 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{21} \left (5 b^3\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx \\ & = \frac {10 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (5 b^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}} \\ & = \frac {10 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {b^2 \sqrt {b \cos (c+d x)} \left (20 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} (23 \sin (c+d x)+3 \sin (3 (c+d x)))\right )}{42 d \sqrt {\cos (c+d x)}} \]
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Time = 3.61 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.16
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{3} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(210\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98 \[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {-5 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (3 \, b^{2} \cos \left (d x + c\right )^{2} + 5 \, b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{21 \, d} \]
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Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
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\[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{5/2} \, dx=\int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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