Integrand size = 43, antiderivative size = 271 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {4 a^3 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (140 A+147 B+106 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d} \]
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Time = 0.69 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4173, 4103, 4082, 3872, 3856, 2719, 2720} \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 a^3 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{105 d}+\frac {2 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {4 a^3 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^3 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{35 a d}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rule 4173
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {1}{2} a (7 A-C)+\frac {1}{2} a (7 B+6 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{7 a} \\ & = \frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (35 A-7 B-11 C)+\frac {7}{4} a^2 (5 A+9 B+7 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{35 a} \\ & = \frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {1}{4} a^3 (35 A-42 B-41 C)+\frac {1}{4} a^3 (140 A+147 B+106 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{105 a} \\ & = \frac {4 a^3 (140 A+147 B+106 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {16 \int \frac {-\frac {21}{8} a^4 (5 A+9 B+7 C)+\frac {5}{8} a^4 (35 A+21 B+13 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{105 a} \\ & = \frac {4 a^3 (140 A+147 B+106 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{5} \left (2 a^3 (5 A+9 B+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (35 A+21 B+13 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {4 a^3 (140 A+147 B+106 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{5} \left (2 a^3 (5 A+9 B+7 C) \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 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {4 a^3 (5 A+9 B+7 C) \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 (35 A+21 B+13 C) \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 (140 A+147 B+106 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.78 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.32 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {a^3 e^{-i d x} \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (-630 i A-1134 i B-882 i C-840 i A \cos (2 (c+d x))-1512 i B \cos (2 (c+d x))-1176 i C \cos (2 (c+d x))-210 i A \cos (4 (c+d x))-378 i B \cos (4 (c+d x))-294 i C \cos (4 (c+d x))+80 (35 A+21 B+13 C) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+14 i (5 A+9 B+7 C) e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+70 A \sin (c+d x)+210 B \sin (c+d x)+380 C \sin (c+d x)+630 A \sin (2 (c+d x))+840 B \sin (2 (c+d x))+840 C \sin (2 (c+d x))+70 A \sin (3 (c+d x))+210 B \sin (3 (c+d x))+260 C \sin (3 (c+d x))+315 A \sin (4 (c+d x))+378 B \sin (4 (c+d x))+294 C \sin (4 (c+d x))\right )}{420 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1071\) vs. \(2(295)=590\).
Time = 7.21 (sec) , antiderivative size = 1072, normalized size of antiderivative = 3.96
method | result | size |
default | \(\text {Expression too large to display}\) | \(1072\) |
parts | \(\text {Expression too large to display}\) | \(1326\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (35 \, A + 21 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (35 \, A + 21 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, A + 9 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (5 \, A + 9 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (21 \, {\left (15 \, A + 18 \, B + 14 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 21 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 21 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 15 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
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