Integrand size = 23, antiderivative size = 147 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {32 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {20 a^2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {20 a^2 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a^2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {4 a^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^2 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2836, 2715, 2719, 2720} \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {20 a^2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {32 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {4 a^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {32 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {20 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2836
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cos ^{\frac {5}{2}}(c+d x)+2 a^2 \cos ^{\frac {7}{2}}(c+d x)+a^2 \cos ^{\frac {9}{2}}(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^{\frac {5}{2}}(c+d x) \, dx+a^2 \int \cos ^{\frac {9}{2}}(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^{\frac {7}{2}}(c+d x) \, dx \\ & = \frac {2 a^2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {4 a^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^2 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{5} \left (3 a^2\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{9} \left (7 a^2\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{7} \left (10 a^2\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {6 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {20 a^2 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a^2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {4 a^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^2 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{15} \left (7 a^2\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (10 a^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {32 a^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {20 a^2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {20 a^2 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {32 a^2 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {4 a^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a^2 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.73 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (\frac {672 (3 \cos (c-d x-\arctan (\tan (c)))+\cos (c+d x+\arctan (\tan (c)))) \csc (c) \sec (c)}{\sqrt {\sec ^2(c)}}-1200 \cos (c+d x) \sqrt {\cos ^2(d x-\arctan (\cot (c)))} \sqrt {\csc ^2(c)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec (d x-\arctan (\cot (c))) \sin (c)+\cos (c+d x) (-2688 \cot (c)+1380 \sin (c+d x)+518 \sin (2 (c+d x))+180 \sin (3 (c+d x))+35 \sin (4 (c+d x)))-1344 \cos (c) \csc (d x+\arctan (\tan (c))) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sqrt {\sec ^2(c)} \sqrt {\sin ^2(d x+\arctan (\tan (c)))}\right )}{5040 d \sqrt {\cos (c+d x)}} \]
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Time = 12.15 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.77
method | result | size |
default | \(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{2} \left (560 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-960 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+608 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-205 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-168 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+93 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(260\) |
parts | \(-\frac {2 a^{2} \sqrt {\left (2 \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 (-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )\right )}{5 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}-\frac {2 a^{2} \sqrt {\left (2 \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 (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 {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}-\frac {4 a^{2} \sqrt {\left (2 \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 (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 {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(621\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.19 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=-\frac {2 \, {\left (75 i \, \sqrt {2} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 75 i \, \sqrt {2} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 168 i \, \sqrt {2} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 168 i \, \sqrt {2} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, a^{2} \cos \left (d x + c\right )^{3} + 90 \, a^{2} \cos \left (d x + c\right )^{2} + 112 \, a^{2} \cos \left (d x + c\right ) + 150 \, a^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \]
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Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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Time = 15.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx=-\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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