Integrand size = 21, antiderivative size = 125 \[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {30 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d \sqrt {b \cos (c+d x)}}+\frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d} \]
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Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2715, 2721, 2720} \[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {30 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d \sqrt {b \cos (c+d x)}}+\frac {2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^2 d}+\frac {30 b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{77 d}+\frac {18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 d} \]
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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))^{11/2} \, dx}{b^3} \\ & = \frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {9 \int (b \cos (c+d x))^{7/2} \, dx}{11 b} \\ & = \frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {1}{77} (45 b) \int (b \cos (c+d x))^{3/2} \, dx \\ & = \frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {1}{77} \left (15 b^3\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx \\ & = \frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d}+\frac {\left (15 b^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 \sqrt {b \cos (c+d x)}} \\ & = \frac {30 b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 d \sqrt {b \cos (c+d x)}}+\frac {30 b^2 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^2 d} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.66 \[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {(b \cos (c+d x))^{5/2} \left (240 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} (290 \sin (c+d x)+57 \sin (3 (c+d x))+7 \sin (5 (c+d x)))\right )}{616 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Time = 10.46 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.89
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{3} \left (448 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1568 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2384 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2040 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1084 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-370 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+62 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(236\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86 \[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\frac {-15 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (7 \, b^{2} \cos \left (d x + c\right )^{4} + 9 \, b^{2} \cos \left (d x + c\right )^{2} + 15 \, b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{77 \, d} \]
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Timed out. \[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(c+d x) (b \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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