Integrand size = 43, antiderivative size = 419 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 b \left (609 a^2 b B+63 b^3 B-a^3 (70 A-366 C)+84 a b^2 (5 A+3 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 b^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \]
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Time = 1.33 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4179, 4181, 4161, 4132, 3856, 2720, 4131, 2719} \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{105 d}+\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (-\left (a^3 (70 A-366 C)\right )+609 a^2 b B+84 a b^2 (5 A+3 C)+63 b^3 B\right )}{105 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{5 d}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} (35 a A-39 a C-21 b B) (a+b \sec (c+d x))^2}{105 d}-\frac {2 b (7 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{21 d}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{3 d \sqrt {\sec (c+d x)}} \]
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Rule 2719
Rule 2720
Rule 3856
Rule 4131
Rule 4132
Rule 4161
Rule 4179
Rule 4181
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2}{3} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} (8 A b+3 a B)+\frac {1}{2} (3 b B+a (A+3 C)) \sec (c+d x)-\frac {1}{2} b (7 A-3 C) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = -\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {4}{21} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {3}{4} a (21 A b+7 a B-b C)+\frac {1}{4} \left (42 a b B+7 a^2 (A+3 C)+3 b^2 (7 A+5 C)\right ) \sec (c+d x)-\frac {1}{4} b (35 a A-21 b B-39 a C) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = -\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {8}{105} \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{8} a \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right )+\frac {1}{8} \left (315 a^2 b B+63 b^3 B+35 a^3 (A+3 C)+3 a b^2 (105 A+59 C)\right ) \sec (c+d x)+\frac {3}{8} b \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 b^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {16}{315} \int \frac {\frac {3}{16} a^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right )+\frac {15}{16} \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \sec (c+d x)+\frac {3}{16} b \left (609 a^2 b B+63 b^3 B-a^3 (70 A-366 C)+84 a b^2 (5 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 b^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {16}{315} \int \frac {\frac {3}{16} a^2 \left (350 a A b+105 a^2 B-21 b^2 B-54 a b C\right )+\frac {3}{16} b \left (609 a^2 b B+63 b^3 B-a^3 (70 A-366 C)+84 a b^2 (5 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 b \left (609 a^2 b B+63 b^3 B-a^3 (70 A-366 C)+84 a b^2 (5 A+3 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 b^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{5} \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (\left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 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 (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 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 (609 a^2 b B+63 b^3 B-a^3 (70 A-366 C)+84 a b^2 (5 A+3 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 b^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{5} \left (\left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 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 (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 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 (609 a^2 b B+63 b^3 B-a^3 (70 A-366 C)+84 a b^2 (5 A+3 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 b^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 b (35 a A-21 b B-39 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d}-\frac {2 b (7 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \\ \end{align*}
Time = 12.70 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (42 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+840 a A b^3 \sin (c+d x)+1260 a^2 b^2 B \sin (c+d x)+126 b^4 B \sin (c+d x)+840 a^3 b C \sin (c+d x)+504 a b^3 C \sin (c+d x)+35 a^4 A \sin (2 (c+d x))+70 A b^4 \tan (c+d x)+280 a b^3 B \tan (c+d x)+420 a^2 b^2 C \tan (c+d x)+50 b^4 C \tan (c+d x)+42 b^4 B \sec (c+d x) \tan (c+d x)+168 a b^3 C \sec (c+d x) \tan (c+d x)+30 b^4 C \sec ^2(c+d x) \tan (c+d x)\right )}{105 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {11}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1537\) vs. \(2(439)=878\).
Time = 8.55 (sec) , antiderivative size = 1538, normalized size of antiderivative = 3.67
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1538\) |
default | \(\text {Expression too large to display}\) | \(1597\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.14 \[ \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 {3}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (A + 3 \, C\right )} a^{4} + 84 i \, B a^{3} b + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + 28 i \, B a b^{3} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \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 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 84 i \, B a^{3} b - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - 28 i \, B a b^{3} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \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 \, \sqrt {2} {\left (-5 i \, B a^{4} - 20 i \, {\left (A - C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} + 3 i \, B b^{4}\right )} \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 \, \sqrt {2} {\left (5 i \, B a^{4} + 20 i \, {\left (A - C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} - 3 i \, B b^{4}\right )} \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 {2 \, {\left (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 15 \, C b^{4} + 21 \, {\left (20 \, C a^{3} b + 30 \, B a^{2} b^{2} + 4 \, {\left (5 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\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 )^{3}} \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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