Integrand size = 43, antiderivative size = 308 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {(9 A-49 B+119 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.73 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4169, 4104, 3872, 3853, 3856, 2719, 2720} \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {(9 A-49 B+119 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{30 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {(3 A-13 B+33 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 a^3 d}-\frac {(9 A-49 B+119 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}+\frac {(9 A-49 B+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A-B+C) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac {(B-2 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3872
Rule 4104
Rule 4169
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (\frac {1}{2} a (3 A+7 B-7 C)+\frac {1}{2} a (3 A-3 B+13 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {25}{2} a^2 (B-2 C)+\frac {3}{2} a^2 (3 A-8 B+23 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \sec ^{\frac {3}{2}}(c+d x) \left (-\frac {3}{4} a^3 (9 A-49 B+119 C)+\frac {15}{4} a^3 (3 A-13 B+33 C) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(3 A-13 B+33 C) \int \sec ^{\frac {5}{2}}(c+d x) \, dx}{4 a^3}-\frac {(9 A-49 B+119 C) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3} \\ & = -\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(3 A-13 B+33 C) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}+\frac {(9 A-49 B+119 C) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = -\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left ((3 A-13 B+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}+\frac {\left ((9 A-49 B+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = \frac {(9 A-49 B+119 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.24 (sec) , antiderivative size = 1462, normalized size of antiderivative = 4.75 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {6 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {98 \sqrt {2} B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{15 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}-\frac {238 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{15 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {4 A \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}-\frac {52 B \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {44 C \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {\cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {4 (9 A-49 B+119 C) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{5 d}+\frac {4 \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{5 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 A \sin \left (\frac {d x}{2}\right )-8 B \sin \left (\frac {d x}{2}\right )+13 C \sin \left (\frac {d x}{2}\right )\right )}{15 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 A \sin \left (\frac {d x}{2}\right )-13 B \sin \left (\frac {d x}{2}\right )+29 C \sin \left (\frac {d x}{2}\right )\right )}{3 d}+\frac {32 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac {8 (4 C+3 A \cos (c)-13 B \cos (c)+33 C \cos (c)) \sec (c) \tan \left (\frac {c}{2}\right )}{3 d}+\frac {8 (3 A-8 B+13 C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{15 d}+\frac {4 (A-B+C) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{5 d}\right )}{(A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1012\) vs. \(2(328)=656\).
Time = 80.37 (sec) , antiderivative size = 1013, normalized size of antiderivative = 3.29
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.93 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {5 \, {\left (\sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, {\left (\sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )\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 ) + 3 \, {\left (\sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )\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 ) + \frac {2 \, {\left (3 \, {\left (9 \, A - 49 \, B + 119 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (33 \, A - 188 \, B + 453 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (9 \, A - 59 \, B + 139 \, C\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left (B - 2 \, C\right )} \cos \left (d x + c\right ) - 20 \, C\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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