Optimal. Leaf size=516 \[ \frac {2 \left (364 a^3 b B+468 a b^3 B+39 b^4 (3 A+5 C)+78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]
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Rubi [A]
time = 0.96, antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4179, 4159,
4132, 3854, 3856, 2720, 4130, 2719} \begin {gather*} \frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right )}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right )}{231 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 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right )}{195 d}+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 4130
Rule 4132
Rule 4159
Rule 4179
Rubi steps
\begin {align*} \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 {13}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2}{13} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} (8 A b+13 a B)+\frac {1}{2} (11 a A+13 b B+13 a C) \sec (c+d x)+\frac {1}{2} b (3 A+13 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {4}{143} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right )+\frac {1}{4} \left (226 a A b+117 a^2 B+143 b^2 B+286 a b C\right ) \sec (c+d x)+\frac {1}{4} b (57 A b+39 a B+143 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{8} \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right )+\frac {1}{8} \left (2951 a^2 b B+1287 b^3 B+77 a^3 (11 A+13 C)+3 a b^2 (961 A+1287 C)\right ) \sec (c+d x)+\frac {3}{8} b \left (338 a b B+11 a^2 (11 A+13 C)+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287}\\ &=\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {7}{16} \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right )-\frac {117}{16} \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sec (c+d x)-\frac {21}{16} b^2 \left (338 a b B+11 a^2 (11 A+13 C)+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{9009}\\ &=\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {7}{16} \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right )-\frac {21}{16} b^2 \left (338 a b B+11 a^2 (11 A+13 C)+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{9009}-\frac {1}{77} \left (-45 a^4 B-330 a^2 b^2 B-77 b^4 B-44 a b^3 (5 A+7 C)-20 a^3 b (9 A+11 C)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}-\frac {1}{231} \left (-45 a^4 B-330 a^2 b^2 B-77 b^4 B-44 a b^3 (5 A+7 C)-20 a^3 b (9 A+11 C)\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{195} \left (-364 a^3 b B-468 a b^3 B-39 b^4 (3 A+5 C)-78 a^2 b^2 (7 A+9 C)-7 a^4 (11 A+13 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}-\frac {1}{231} \left (\left (-45 a^4 B-330 a^2 b^2 B-77 b^4 B-44 a b^3 (5 A+7 C)-20 a^3 b (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{195} \left (\left (-364 a^3 b B-468 a b^3 B-39 b^4 (3 A+5 C)-78 a^2 b^2 (7 A+9 C)-7 a^4 (11 A+13 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (364 a^3 b B+468 a b^3 B+39 b^4 (3 A+5 C)+78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 7.30, size = 658, normalized size = 1.28 \begin {gather*} \frac {2 \cos ^6(c+d x) \left (\frac {2 \left (5929 a^4 A+42042 a^2 A b^2+9009 A b^4+28028 a^3 b B+36036 a b^3 B+7007 a^4 C+54054 a^2 b^2 C+15015 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 (11700 a^3 A b+14300 a A b^3+2925 a^4 B+21450 a^2 b^2 B+5005 b^4 B+14300 a^3 b C+20020 a b^3 C\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |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 )}{15015 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 {\left (1897 a^4 A+11856 a^2 A b^2+1872 A b^4+7904 a^3 b B+7488 a b^3 B+1976 a^4 C+11232 a^2 b^2 C\right ) \sin (c+d x)}{9360}+\frac {\left (4164 a^3 A b+4576 a A b^3+1041 a^4 B+6864 a^2 b^2 B+1232 b^4 B+4576 a^3 b C+4928 a b^3 C\right ) \sin (2 (c+d x))}{1848}+\frac {\left (2297 a^4 A+13416 a^2 A b^2+1872 A b^4+8944 a^3 b B+7488 a b^3 B+2236 a^4 C+11232 a^2 b^2 C\right ) \sin (3 (c+d x))}{9360}+\frac {1}{77} a \left (32 a^2 A b+22 A b^3+8 a^3 B+33 a b^2 B+22 a^2 b C\right ) \sin (4 (c+d x))+\frac {a^2 \left (89 a^2 A+312 A b^2+208 a b B+52 a^2 C\right ) \sin (5 (c+d x))}{1872}+\frac {1}{88} a^3 (4 A b+a B) \sin (6 (c+d x))+\frac {1}{208} a^4 A \sin (7 (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)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1406\) vs.
\(2(532)=1064\).
time = 0.13, size = 1407, normalized size = 2.73
method | result | size |
default | \(\text {Expression too large to display}\) | \(1407\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.12, size = 560, normalized size = 1.09 \begin {gather*} -\frac {195 \, \sqrt {2} {\left (45 i \, B a^{4} + 20 i \, {\left (9 \, A + 11 \, C\right )} a^{3} b + 330 i \, B a^{2} b^{2} + 44 i \, {\left (5 \, A + 7 \, C\right )} a b^{3} + 77 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (-45 i \, B a^{4} - 20 i \, {\left (9 \, A + 11 \, C\right )} a^{3} b - 330 i \, B a^{2} b^{2} - 44 i \, {\left (5 \, A + 7 \, C\right )} a b^{3} - 77 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, {\left (11 \, A + 13 \, C\right )} a^{4} - 364 i \, B a^{3} b - 78 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b^{2} - 468 i \, B a b^{3} - 39 i \, {\left (3 \, A + 5 \, 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 ) + 231 \, \sqrt {2} {\left (7 i \, {\left (11 \, A + 13 \, C\right )} a^{4} + 364 i \, B a^{3} b + 78 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b^{2} + 468 i \, B a b^{3} + 39 i \, {\left (3 \, A + 5 \, 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 (3465 \, A a^{4} \cos \left (d x + c\right )^{6} + 4095 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{5} + 385 \, {\left ({\left (11 \, A + 13 \, C\right )} a^{4} + 52 \, B a^{3} b + 78 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 585 \, {\left (9 \, B a^{4} + 4 \, {\left (9 \, A + 11 \, C\right )} a^{3} b + 66 \, B a^{2} b^{2} + 44 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, {\left (11 \, A + 13 \, C\right )} a^{4} + 364 \, B a^{3} b + 78 \, {\left (7 \, A + 9 \, C\right )} a^{2} b^{2} + 468 \, B a b^{3} + 117 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 195 \, {\left (45 \, B a^{4} + 20 \, {\left (9 \, A + 11 \, C\right )} a^{3} b + 330 \, B a^{2} b^{2} + 44 \, {\left (5 \, A + 7 \, C\right )} a b^{3} + 77 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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