Integrand size = 43, antiderivative size = 219 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {4 a^2 (3 A+4 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^2 (6 A+7 B+14 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (4 A+7 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.57 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4171, 4102, 4081, 3872, 3856, 2719, 2720} \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a^2 (6 A+7 B+14 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^2 (3 A+4 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (4 A+7 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 3872
Rule 4081
Rule 4102
Rule 4171
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{2} a (4 A+7 B)+\frac {1}{2} a (A+7 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{7 a} \\ & = \frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (4 A+7 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x)) \left (\frac {1}{4} a^2 (33 A+49 B+35 C)+\frac {1}{4} a^2 (9 A+7 B+35 C) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{35 a} \\ & = \frac {2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (4 A+7 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {-\frac {21}{4} a^3 (3 A+4 B+5 C)-\frac {5}{4} a^3 (6 A+7 B+14 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{105 a} \\ & = \frac {2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (4 A+7 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (2 a^2 (3 A+4 B+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^2 (6 A+7 B+14 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (4 A+7 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (2 a^2 (3 A+4 B+5 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^2 (6 A+7 B+14 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {4 a^2 (3 A+4 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 d}+\frac {4 a^2 (6 A+7 B+14 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 {2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (4 A+7 B) \left (a^2+a^2 \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 = 3.48 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.86 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {a^2 \sqrt {\sec (c+d x)} \left (80 (6 A+7 (B+2 C)) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-112 i (3 A+4 B+5 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+2 \cos (c+d x) (504 i A+672 i B+840 i C+5 (51 A+28 (2 B+C)) \sin (c+d x)+42 (2 A+B) \sin (2 (c+d x))+15 A \sin (3 (c+d x)))\right )}{420 d} \]
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Time = 7.21 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.21
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^{2} \left (120 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-348 A -84 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (378 A +224 B +70 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-117 A -91 B -35 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+30 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-63 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+35 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-84 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+70 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-105 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \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}\) | \(483\) |
parts | \(\text {Expression too large to display}\) | \(892\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.03 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (6 \, A + 7 \, B + 14 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (6 \, A + 7 \, B + 14 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (3 \, A + 4 \, B + 5 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (3 \, A + 4 \, B + 5 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (15 \, A a^{2} \cos \left (d x + c\right )^{3} + 21 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{2} + 5 \, {\left (12 \, A + 14 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d} \]
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\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=a^{2} \left (\int \frac {A}{\sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 A}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 B}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {B}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {C}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {2 C}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int C \sqrt {\sec {\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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