Optimal. Leaf size=401 \[ \frac {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A]
time = 0.61, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps
used = 10, 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 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 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 (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 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 (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{15 d}+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{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))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {1}{2} (6 A b+11 a B)+\frac {1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)+\frac {1}{2} b (3 A+11 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4}{99} \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{4} \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right )+\frac {1}{4} \left (150 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \sec (c+d x)+\frac {3}{4} b (15 A b+11 a B+33 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {8}{693} \int \frac {-\frac {7}{8} \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right )-\frac {9}{8} \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sec (c+d x)-\frac {21}{8} b^2 (15 A b+11 a B+33 b C) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {8}{693} \int \frac {-\frac {7}{8} \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right )-\frac {21}{8} b^2 (15 A b+11 a B+33 b C) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx-\frac {1}{77} \left (-165 a^2 b B-77 b^3 B-33 a b^2 (5 A+7 C)-5 a^3 (9 A+11 C)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {1}{15} \left (-7 a^3 B-27 a b^2 B-3 b^3 (3 A+5 C)-3 a^2 b (7 A+9 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{231} \left (-165 a^2 b B-77 b^3 B-33 a b^2 (5 A+7 C)-5 a^3 (9 A+11 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {1}{15} \left (\left (-7 a^3 B-27 a b^2 B-3 b^3 (3 A+5 C)-3 a^2 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx-\frac {1}{231} \left (\left (-165 a^2 b B-77 b^3 B-33 a b^2 (5 A+7 C)-5 a^3 (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 {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (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 (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (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 (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 6.87, size = 538, normalized size = 1.34 \begin {gather*} \frac {2 \cos ^5(c+d x) \left (\frac {2 \left (1617 a^2 A b+693 A b^3+539 a^3 B+2079 a b^2 B+2079 a^2 b C+1155 b^3 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 (225 a^3 A+825 a A b^2+825 a^2 b B+385 b^3 B+275 a^3 C+1155 a b^2 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))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{1155 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{90} \left (57 a^2 A b+18 A b^3+19 a^3 B+54 a b^2 B+54 a^2 b C\right ) \sin (c+d x)+\frac {\left (1041 a^3 A+3432 a A b^2+3432 a^2 b B+1232 b^3 B+1144 a^3 C+3696 a b^2 C\right ) \sin (2 (c+d x))}{1848}+\frac {1}{180} \left (129 a^2 A b+36 A b^3+43 a^3 B+108 a b^2 B+108 a^2 b C\right ) \sin (3 (c+d x))+\frac {1}{154} a \left (16 a^2 A+33 A b^2+33 a b B+11 a^2 C\right ) \sin (4 (c+d x))+\frac {1}{36} a^2 (3 A b+a B) \sin (5 (c+d x))+\frac {1}{88} a^3 A \sin (6 (c+d x))\right )}{d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{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. \(1081\) vs.
\(2(421)=842\).
time = 0.12, size = 1082, normalized size = 2.70
method | result | size |
default | \(\text {Expression too large to display}\) | \(1082\) |
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 = 440, normalized size = 1.10 \begin {gather*} -\frac {15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{3} + 165 i \, B a^{2} b + 33 i \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 77 i \, B b^{3}\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 \, {\left (9 \, A + 11 \, C\right )} a^{3} - 165 i \, B a^{2} b - 33 i \, {\left (5 \, A + 7 \, C\right )} a b^{2} - 77 i \, B b^{3}\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 \, B a^{3} - 3 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b - 27 i \, B a b^{2} - 3 i \, {\left (3 \, A + 5 \, C\right )} b^{3}\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 \, B a^{3} + 3 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b + 27 i \, B a b^{2} + 3 i \, {\left (3 \, A + 5 \, C\right )} b^{3}\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 (315 \, A a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{3} + 33 \, B a^{2} b + 33 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, B a^{3} + 3 \, {\left (7 \, A + 9 \, C\right )} a^{2} b + 27 \, B a b^{2} + 9 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{3} + 165 \, B a^{2} b + 33 \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 77 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, 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 )}^3\,\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 )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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