Integrand size = 43, antiderivative size = 361 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d} \]
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Time = 1.07 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4197, 3128, 3112, 3102, 2827, 2719, 2715, 2720} \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{693 d}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{15 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{495 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{99 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3102
Rule 3112
Rule 3128
Rule 4197
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2}{11} \int \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \left (\frac {1}{2} b (3 A+11 C)+\frac {1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac {1}{2} (6 A b+11 a B) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4}{99} \int \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \left (\frac {3}{4} b (15 A b+11 a B+33 b C)+\frac {1}{4} \left (150 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac {1}{4} \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {8}{693} \int \sqrt {\cos (c+d x)} \left (\frac {21}{8} b^2 (15 A b+11 a B+33 b C)+\frac {9}{8} \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \cos (c+d x)+\frac {7}{8} \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {16 \int \sqrt {\cos (c+d x)} \left (\frac {231}{16} \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right )+\frac {45}{16} \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \cos (c+d x)\right ) \, dx}{3465} \\ & = \frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {1}{15} \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {1}{231} \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d} \\ \end{align*}
Time = 4.43 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.79 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {154 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (154 \left (36 A b^3+43 a^3 B+108 a b^2 B+3 a^2 b (43 A+36 C)\right ) \cos (c+d x)+5 \left (36 a \left (33 A b^2+33 a b B+a^2 (16 A+11 C)\right ) \cos (2 (c+d x))+154 a^2 (3 A b+a B) \cos (3 (c+d x))+3 \left (1716 a^2 b B+616 b^3 B+132 a b^2 (13 A+14 C)+a^3 (531 A+572 C)+21 a^3 A \cos (4 (c+d x))\right )\right )\right ) \sin (c+d x)}{1155 d} \]
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Timed out.
hanged
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.19 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, A a^{3} \cos \left (d x + c\right )^{4} + 75 \, {\left (9 \, A + 11 \, C\right )} a^{3} + 2475 \, B a^{2} b + 495 \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 1155 \, B b^{3} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{3} + 33 \, B a^{2} b + 33 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (7 \, B a^{3} + 3 \, {\left (7 \, A + 9 \, C\right )} a^{2} b + 27 \, B a b^{2} + 9 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{3} + 165 i \, B a^{2} b + 33 i \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 77 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-5 i \, {\left (9 \, A + 11 \, C\right )} a^{3} - 165 i \, B a^{2} b - 33 i \, {\left (5 \, A + 7 \, C\right )} a b^{2} - 77 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-7 i \, B a^{3} - 3 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b - 27 i \, B a b^{2} - 3 i \, {\left (3 \, A + 5 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (7 i \, B a^{3} + 3 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b + 27 i \, B a b^{2} + 3 i \, {\left (3 \, A + 5 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \]
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Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
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Time = 20.68 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.42 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (C\,b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {B\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^3\,{\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 {2\,B\,a^3\,{\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}}-\frac {2\,C\,a^3\,{\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\,a\,b^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 )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a^2\,b\,{\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}}-\frac {6\,B\,a\,b^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 {2\,B\,a^2\,b\,{\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 )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^2\,b\,{\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}} \]
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