Optimal. Leaf size=401 \[ \frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{15 d}+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 0.91, 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 = {4094, 4074, 4047, 3769, 3771, 2641, 4045, 2639} \[ \frac {2 \sin (c+d x) \left (33 a^2 b (7 A+9 C)+77 a^3 B+242 a b^2 B+24 A b^3\right )}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (7 A+9 C)+7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{15 d}+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 4045
Rule 4047
Rule 4074
Rule 4094
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 \[ \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{88} a^3 A \sin (6 (c+d x))+\frac {1}{154} a \sin (4 (c+d x)) \left (16 a^2 A+11 a^2 C+33 a b B+33 A b^2\right )+\frac {1}{36} a^2 (a B+3 A b) \sin (5 (c+d x))+\frac {1}{90} \sin (c+d x) \left (19 a^3 B+57 a^2 A b+54 a^2 b C+54 a b^2 B+18 A b^3\right )+\frac {\sin (2 (c+d x)) \left (1041 a^3 A+1144 a^3 C+3432 a^2 b B+3432 a A b^2+3696 a b^2 C+1232 b^3 B\right )}{1848}+\frac {1}{180} \sin (3 (c+d x)) \left (43 a^3 B+129 a^2 A b+108 a^2 b C+108 a b^2 B+36 A b^3\right )\right )}{d \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac {2 \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (225 a^3 A+275 a^3 C+825 a^2 b B+825 a A b^2+1155 a b^2 C+385 b^3 B\right )+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (539 a^3 B+1617 a^2 A b+2079 a^2 b C+2079 a b^2 B+693 A b^3+1155 b^3 C\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}\right )}{1155 d (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{3} \sec \left (d x + c\right )^{5} + {\left (3 \, C a b^{2} + B b^{3}\right )} \sec \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.80, size = 1082, normalized size = 2.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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