Integrand size = 43, antiderivative size = 429 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Time = 1.40 (sec) , antiderivative size = 429, 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 = {4179, 4161, 4132, 3856, 2720, 4131, 2719} \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{105 d}+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{105 d \sqrt {\sec (c+d x)}}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right )}{105 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{5 d}+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 4131
Rule 4132
Rule 4161
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2}{7} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} (8 A b+7 a B)+\frac {1}{2} (5 a A+7 b B+7 a C) \sec (c+d x)-\frac {1}{2} b (3 A-7 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4}{35} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right )+\frac {1}{4} \left (34 a A b+21 a^2 B+35 b^2 B+70 a b C\right ) \sec (c+d x)-\frac {1}{4} b (39 A b+21 a B-35 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8}{105} \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{8} \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right )+\frac {1}{8} \left (77 a^2 b B+105 b^3 B+5 a^3 (5 A+7 C)+3 a b^2 (11 A+105 C)\right ) \sec (c+d x)-\frac {3}{8} b \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = -\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16}{315} \int \frac {\frac {3}{16} a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right )+\frac {15}{16} \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sec (c+d x)-\frac {3}{16} b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx \\ & = -\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16}{315} \int \frac {\frac {3}{16} a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right )-\frac {3}{16} b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = -\frac {2 b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (\left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \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 b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \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 {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \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 b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \\ \end{align*}
Time = 12.88 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (168 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+168 a^3 A b \sin (c+d x)+42 a^4 B \sin (c+d x)+840 b^4 B \sin (c+d x)+3360 a b^3 C \sin (c+d x)+130 a^4 A \sin (2 (c+d x))+840 a^2 A b^2 \sin (2 (c+d x))+560 a^3 b B \sin (2 (c+d x))+140 a^4 C \sin (2 (c+d x))+168 a^3 A b \sin (3 (c+d x))+42 a^4 B \sin (3 (c+d x))+15 a^4 A \sin (4 (c+d x))+280 b^4 C \tan (c+d x)\right )}{210 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {11}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1337\) vs. \(2(449)=898\).
Time = 9.07 (sec) , antiderivative size = 1338, normalized size of antiderivative = 3.12
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1338\) |
default | \(\text {Expression too large to display}\) | \(2507\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{4} + 28 i \, B a^{3} b + 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} + 84 i \, B a b^{3} + 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{4} - 28 i \, B a^{3} b - 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} - 84 i \, B a b^{3} - 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{4} - 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 20 i \, {\left (A - C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\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 ) + 21 \, \sqrt {2} {\left (3 i \, B a^{4} + 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 20 i \, {\left (A - C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\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 (15 \, A a^{4} \cos \left (d x + c\right )^{4} + 35 \, C b^{4} + 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )} \]
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\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{4} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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