Integrand size = 35, antiderivative size = 319 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {4 a^3 (221 A+175 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]
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Time = 0.84 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4306, 3125, 3055, 3047, 3102, 2827, 2715, 2720, 2719} \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (143 A+145 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (221 A+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {12 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rule 3125
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+5 C)+3 a C \cos (c+d x)\right ) \, dx}{13 a} \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (\frac {1}{4} a^2 (143 A+85 C)+\frac {1}{4} a^2 (143 A+145 C) \cos (c+d x)\right ) \, dx}{143 a} \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x)) \left (\frac {1}{4} a^3 (1001 A+745 C)+\frac {5}{2} a^3 (143 A+118 C) \cos (c+d x)\right ) \, dx}{1287 a} \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{4} a^4 (1001 A+745 C)+\left (\frac {5}{2} a^4 (143 A+118 C)+\frac {1}{4} a^4 (1001 A+745 C)\right ) \cos (c+d x)+\frac {5}{2} a^4 (143 A+118 C) \cos ^2(c+d x)\right ) \, dx}{1287 a} \\ & = \frac {40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {117}{8} a^4 (121 A+95 C)+\frac {77}{8} a^4 (221 A+175 C) \cos (c+d x)\right ) \, dx}{9009 a} \\ & = \frac {40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{77} \left (2 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{117} \left (2 a^3 (221 A+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (2 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{195} \left (2 a^3 (221 A+175 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {4 a^3 (221 A+175 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (143 A+145 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.08 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.78 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (12480 (121 A+95 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-4928 i (221 A+175 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (3267264 i A+2587200 i C+780 (2134 A+1811 C) \sin (c+d x)+77 (7592 A+7825 C) \sin (2 (c+d x))+154440 A \sin (3 (c+d x))+251550 C \sin (3 (c+d x))+20020 A \sin (4 (c+d x))+90860 C \sin (4 (c+d x))+24570 C \sin (5 (c+d x))+3465 C \sin (6 (c+d x)))\right )}{720720 d} \]
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Time = 15.47 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.45
method | result | size |
default | \(-\frac {4 \sqrt {\left (-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (-221760 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1058400 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-80080 A -2122400 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (314600 A +2331040 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-487916 A -1535860 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (386386 A +633710 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-105534 A -121230 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+23595 A \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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 A \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 )+18525 C \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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 C \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 )}{45045 \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 {-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(464\) |
parts | \(\text {Expression too large to display}\) | \(1291\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.84 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (195 i \, \sqrt {2} {\left (121 \, A + 95 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {2} {\left (121 \, A + 95 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (221 \, A + 175 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (221 \, A + 175 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (3465 \, C a^{3} \cos \left (d x + c\right )^{6} + 12285 \, C a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (13 \, A + 50 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 585 \, {\left (33 \, A + 38 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 154 \, {\left (221 \, A + 175 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 390 \, {\left (121 \, A + 95 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{45045 \, d} \]
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\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=a^{3} \left (\int \frac {A}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A \cos {\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C \cos ^{3}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 C \cos ^{4}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \cos ^{5}{\left (c + d x \right )}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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