Integrand size = 23, antiderivative size = 138 \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {47 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}+\frac {59 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {59 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}+\frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d} \]
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Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2872, 3102, 2832, 2831, 2740, 2732} \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {59 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {47 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}+\frac {\sin (c+d x) \cos (c+d x) (4 \cos (c+d x)+3)^{3/2}}{14 d}-\frac {3 \sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{70 d}+\frac {59 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{105 d} \]
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Rule 2732
Rule 2740
Rule 2831
Rule 2832
Rule 2872
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d}+\frac {1}{14} \int \sqrt {3+4 \cos (c+d x)} \left (3+10 \cos (c+d x)-6 \cos ^2(c+d x)\right ) \, dx \\ & = -\frac {3 (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}+\frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d}+\frac {1}{140} \int \sqrt {3+4 \cos (c+d x)} (-6+118 \cos (c+d x)) \, dx \\ & = \frac {59 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}+\frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d}+\frac {1}{210} \int \frac {209+141 \cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx \\ & = \frac {59 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}+\frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d}+\frac {47}{280} \int \sqrt {3+4 \cos (c+d x)} \, dx+\frac {59}{120} \int \frac {1}{\sqrt {3+4 \cos (c+d x)}} \, dx \\ & = \frac {47 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}+\frac {59 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {59 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}+\frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.67 \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {141 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+59 \sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )+\sqrt {3+4 \cos (c+d x)} (212 \sin (c+d x)+9 \sin (2 (c+d x))+30 \sin (3 (c+d x)))}{420 d} \]
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Time = 6.00 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99
method | result | size |
default | \(-\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (7680 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14976 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12344 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4480 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+413 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-141 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{420 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(275\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07 \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {4 \, {\left (60 \, \cos \left (d x + c\right )^{2} + 9 \, \cos \left (d x + c\right ) + 91\right )} \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) - 277 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 277 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 282 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 282 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{840 \, d} \]
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Timed out. \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^3\,\sqrt {4\,\cos \left (c+d\,x\right )+3} \,d x \]
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