Integrand size = 43, antiderivative size = 426 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (36 a^3 b B+60 a b^3 B+15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
[Out]
Time = 1.38 (sec) , antiderivative size = 426, 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, 4159, 4132, 3856, 2720, 4131, 2719} \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{315 d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right )}{15 d}+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
[In]
[Out]
Rule 2719
Rule 2720
Rule 3856
Rule 4131
Rule 4132
Rule 4159
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} (8 A b+9 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)-\frac {1}{2} b (A-9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{63} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right )+\frac {1}{4} \left (82 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)-\frac {3}{4} b (5 A b+3 a B-21 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {8}{315} \int \frac {(a+b \sec (c+d x)) \left (\frac {3}{8} \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right )+\frac {1}{8} \left (531 a^2 b B+315 b^3 B+21 a^3 (7 A+9 C)+a b^2 (479 A+945 C)\right ) \sec (c+d x)-\frac {1}{8} b \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {16}{945} \int \frac {-\frac {3}{16} \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )-\frac {45}{16} \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sec (c+d x)+\frac {3}{16} b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {16}{945} \int \frac {-\frac {3}{16} \left (192 A b^4+756 a^3 b B+1098 a b^3 B+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right )+\frac {3}{16} b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{21} \left (-5 a^4 B-42 a^2 b^2 B-21 b^4 B-28 a b^3 (A+3 C)-4 a^3 b (5 A+7 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {1}{15} \left (-36 a^3 b B-60 a b^3 B-15 b^4 (A-C)-18 a^2 b^2 (3 A+5 C)-a^4 (7 A+9 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{21} \left (\left (-5 a^4 B-42 a^2 b^2 B-21 b^4 B-28 a b^3 (A+3 C)-4 a^3 b (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 (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (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 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {1}{15} \left (\left (-36 a^3 b B-60 a b^3 B-15 b^4 (A-C)-18 a^2 b^2 (3 A+5 C)-a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 \left (36 a^3 b B+60 a b^3 B+15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (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 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Time = 13.77 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (49 a^4 A+378 a^2 A b^2+105 A b^4+252 a^3 b B+420 a b^3 B+63 a^4 C+630 a^2 b^2 C-105 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (100 a^3 A b+140 a A b^3+25 a^4 B+210 a^2 b^2 B+105 b^4 B+140 a^3 b C+420 a b^3 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{105 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{90} \left (19 a^4 A+108 a^2 A b^2+72 a^3 b B+18 a^4 C+360 b^4 C\right ) \sin (c+d x)+\frac {1}{21} a \left (52 a^2 A b+56 A b^3+13 a^3 B+84 a b^2 B+56 a^2 b C\right ) \sin (2 (c+d x))+\frac {1}{180} a^2 \left (43 a^2 A+216 A b^2+144 a b B+36 a^2 C\right ) \sin (3 (c+d x))+\frac {1}{14} a^3 (4 A b+a B) \sin (4 (c+d x))+\frac {1}{36} a^4 A \sin (5 (c+d x))\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1336\) vs. \(2(446)=892\).
Time = 12.56 (sec) , antiderivative size = 1337, normalized size of antiderivative = 3.14
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1337\) |
default | \(\text {Expression too large to display}\) | \(1652\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 442, 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 {9}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{4} + 4 i \, {\left (5 \, A + 7 \, C\right )} a^{3} b + 42 i \, B a^{2} b^{2} + 28 i \, {\left (A + 3 \, C\right )} a b^{3} + 21 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{4} - 4 i \, {\left (5 \, A + 7 \, C\right )} a^{3} b - 42 i \, B a^{2} b^{2} - 28 i \, {\left (A + 3 \, C\right )} a b^{3} - 21 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{4} - 36 i \, B a^{3} b - 18 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} - 60 i \, B a b^{3} - 15 i \, {\left (A - C\right )} b^{4}\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 (i \, {\left (7 \, A + 9 \, C\right )} a^{4} + 36 i \, B a^{3} b + 18 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} + 60 i \, B a b^{3} + 15 i \, {\left (A - C\right )} b^{4}\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 (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 315 \, C b^{4} + 45 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{4} + 36 \, B a^{3} b + 54 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 7 \, C\right )} a^{3} b + 42 \, B a^{2} b^{2} + 28 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
[In]
[Out]