Integrand size = 43, antiderivative size = 250 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=-\frac {3 (7 A-7 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (9 A-7 B+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 a d}+\frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \]
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Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4169, 3872, 3854, 3856, 2720, 2719} \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {5 (9 A-7 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a d}-\frac {3 (7 A-7 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d} \]
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 3872
Rule 4169
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {\int \frac {\frac {1}{2} a (9 A-7 B+7 C)-\frac {1}{2} a (7 A-7 B+5 C) \sec (c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{a^2} \\ & = -\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}-\frac {(7 A-7 B+5 C) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{2 a}+\frac {(9 A-7 B+7 C) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{2 a} \\ & = \frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}-\frac {(3 (7 A-7 B+5 C)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{10 a}+\frac {(5 (9 A-7 B+7 C)) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{14 a} \\ & = \frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {(5 (9 A-7 B+7 C)) \int \sqrt {\sec (c+d x)} \, dx}{42 a}-\frac {\left (3 (7 A-7 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{10 a} \\ & = -\frac {3 (7 A-7 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))}+\frac {\left (5 (9 A-7 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{42 a} \\ & = -\frac {3 (7 A-7 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (9 A-7 B+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 a d}+\frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.45 (sec) , antiderivative size = 1406, normalized size of antiderivative = 5.62 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\frac {7 \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 ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos (c+d x) \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 ) \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))}-\frac {7 \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 ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos (c+d x) \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 ) \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))}+\frac {\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 ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos (c+d x) \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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))}+\frac {30 A \cos ^2\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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{7 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))}-\frac {10 B \cos ^2\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 ) \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)) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))}+\frac {10 C \cos ^2\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 ) \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)) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))}+\frac {\cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {(51 A-51 B+40 C+33 A \cos (2 c)-33 B \cos (2 c)+20 C \cos (2 c)) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{10 d}+\frac {2 (27 A-14 B+14 C) \cos (2 d x) \sin (2 c)}{21 d}-\frac {2 (A-B) \cos (3 d x) \sin (3 c)}{5 d}+\frac {A \cos (4 d x) \sin (4 c)}{7 d}-\frac {4 \sec \left (\frac {c}{2}\right ) \sec \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 )}{d}-\frac {2 (33 A-33 B+20 C) \cos (c) \sin (d x)}{5 d}+\frac {2 (27 A-14 B+14 C) \cos (2 c) \sin (2 d x)}{21 d}-\frac {2 (A-B) \cos (3 c) \sin (3 d x)}{5 d}+\frac {A \cos (4 c) \sin (4 d x)}{7 d}-\frac {4 (A-B+C) \tan \left (\frac {c}{2}\right )}{d}\right )}{(A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {\sec (c+d x)} (a+a \sec (c+d x))} \]
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Time = 4.31 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (225 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+441 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-175 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-441 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+175 C \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+315 C \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )-480 A \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (864 A +336 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-888 A -392 B -280 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (930 A -210 B +630 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-321 A +161 B -245 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{105 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(341\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=-\frac {25 \, {\left (\sqrt {2} {\left (9 i \, A - 7 i \, B + 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (9 i \, A - 7 i \, B + 7 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 25 \, {\left (\sqrt {2} {\left (-9 i \, A + 7 i \, B - 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-9 i \, A + 7 i \, B - 7 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 \, {\left (\sqrt {2} {\left (7 i \, A - 7 i \, B + 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (7 i \, A - 7 i \, B + 5 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 63 \, {\left (\sqrt {2} {\left (-7 i \, A + 7 i \, B - 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-7 i \, A + 7 i \, B - 5 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (30 \, A \cos \left (d x + c\right )^{4} - 6 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (39 \, A - 14 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{2} + 25 \, {\left (9 \, A - 7 \, B + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{210 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
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\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\frac {\int \frac {A}{\sec ^{\frac {9}{2}}{\left (c + d x \right )} + \sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B \sec {\left (c + d x \right )}}{\sec ^{\frac {9}{2}}{\left (c + d x \right )} + \sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx + \int \frac {C \sec ^{2}{\left (c + d x \right )}}{\sec ^{\frac {9}{2}}{\left (c + d x \right )} + \sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx}{a} \]
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\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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