Integrand size = 33, antiderivative size = 211 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=\frac {4 a^3 (9 A+7 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (21 A+13 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.54 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3039, 4102, 4081, 3872, 3856, 2719, 2720} \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=\frac {2 (7 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (21 A+13 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (9 A+7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 3039
Rule 3856
Rule 3872
Rule 4081
Rule 4102
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+a \sec (c+d x))^3 (B+A \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2}{7} \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{2} a (7 A+11 B)+\frac {1}{2} a (7 A+B) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4}{35} \int \frac {(a+a \sec (c+d x)) \left (\frac {1}{2} a^2 (42 A+41 B)+\frac {1}{2} a^2 (21 A+8 B) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {8}{105} \int \frac {-\frac {21}{4} a^3 (9 A+7 B)-\frac {5}{4} a^3 (21 A+13 B) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (2 a^3 (9 A+7 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (21 A+13 B)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (2 a^3 (9 A+7 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (2 a^3 (21 A+13 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 (9 A+7 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (21 A+13 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (7 A+11 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.22 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.92 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (40 (21 A+13 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-56 i (9 A+7 B) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (168 i (9 A+7 B)+5 (84 A+107 B) \sin (c+d x)+42 (A+3 B) \sin (2 (c+d x))+15 B \sin (3 (c+d x)))\right )}{210 d} \]
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Time = 14.00 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.82
method | result | size |
default | \(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (120 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-84 A -432 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (294 A +602 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-126 A -208 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+105 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-189 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+65 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(385\) |
parts | \(\text {Expression too large to display}\) | \(888\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (21 \, A + 13 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (21 \, A + 13 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (9 \, A + 7 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (9 \, A + 7 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (15 \, B a^{3} \cos \left (d x + c\right )^{3} + 21 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (21 \, A + 26 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d} \]
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\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=a^{3} \left (\int A \sqrt {\sec {\left (c + d x \right )}}\, dx + \int 3 A \cos {\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx + \int 3 A \cos ^{2}{\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx + \int B \cos {\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sqrt {\sec {\left (c + d x \right )}}\, dx\right ) \]
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\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt {\sec (c+d x)} \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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