Integrand size = 33, antiderivative size = 277 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {4 a^3 (15 A+17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (105 A+121 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)} \]
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Time = 0.56 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4102, 4081, 3872, 3854, 3856, 2719, 2720} \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (15 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (105 A+121 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (15 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 3872
Rule 4081
Rule 4102
Rubi steps \begin{align*} \text {integral}& = \frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{2} a (15 A+11 B)+\frac {1}{2} a (5 A+11 B) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{99} \int \frac {(a+a \sec (c+d x)) \left (\frac {5}{2} a^2 (21 A+22 B)+\frac {1}{2} a^2 (60 A+77 B) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {8}{693} \int \frac {-\frac {77}{4} a^3 (15 A+17 B)-\frac {9}{4} a^3 (105 A+121 B) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{9} \left (2 a^3 (15 A+17 B)\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+\frac {1}{77} \left (2 a^3 (105 A+121 B)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{15} \left (2 a^3 (15 A+17 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{231} \left (2 a^3 (105 A+121 B)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{15} \left (2 a^3 (15 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (105 A+121 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {4 a^3 (15 A+17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (105 A+121 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (15 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.18 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (480 (105 A+121 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-2464 i (15 A+17 B) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (110880 i A+125664 i B+30 (1953 A+2134 B) \sin (c+d x)+308 (75 A+73 B) \sin (2 (c+d x))+8505 A \sin (3 (c+d x))+5940 B \sin (3 (c+d x))+2310 A \sin (4 (c+d x))+770 B \sin (4 (c+d x))+315 A \sin (5 (c+d x)))\right )}{27720 d} \]
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Time = 30.20 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.59
method | result | size |
default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (10080 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-43680 A -6160 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (77280 A +24200 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-72240 A -37532 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (39270 A +29722 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-8820 A -8118 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1575 A \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3465 A \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1815 B \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3927 B \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(441\) |
parts | \(\text {Expression too large to display}\) | \(1059\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.91 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (105 \, A + 121 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (105 \, A + 121 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (315 \, A a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 135 \, {\left (14 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 154 \, {\left (15 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \, {\left (105 \, A + 121 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
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