Optimal. Leaf size=345 \[ \frac {2 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A] time = 0.57, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4025, 4074, 4047, 3769, 3771, 2639, 4045, 2641} \[ \frac {2 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (21 a^2 A b+7 a^3 B+27 a b^2 B+9 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 4025
Rule 4045
Rule 4047
Rule 4074
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {2}{11} \int \frac {(a+b \sec (c+d x)) \left (-\frac {1}{2} a (15 A b+11 a B)-\frac {1}{2} \left (9 a^2 A+11 A b^2+22 a b B\right ) \sec (c+d x)-\frac {1}{2} b (5 a A+11 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4}{99} \int \frac {\frac {9}{4} a \left (9 a^2 A+26 A b^2+33 a b B\right )+\frac {11}{4} \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sec (c+d x)+\frac {9}{4} b^2 (5 a A+11 b B) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4}{99} \int \frac {\frac {9}{4} a \left (9 a^2 A+26 A b^2+33 a b B\right )+\frac {9}{4} b^2 (5 a A+11 b B) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx+\frac {1}{9} \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {1}{15} \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{77} \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {1}{231} \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (\left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\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 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {1}{231} \left (\left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\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 (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\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^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 3.23, size = 256, normalized size = 0.74 \[ \frac {\sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) \left (180 a \left (16 a^2 A+33 a b B+33 A b^2\right ) \cos (2 (c+d x))+770 a^2 (a B+3 A b) \cos (3 (c+d x))+154 \left (43 a^3 B+129 a^2 A b+108 a b^2 B+36 A b^3\right ) \cos (c+d x)+15 \left (21 a^3 A \cos (4 (c+d x))+531 a^3 A+1716 a^2 b B+1716 a A b^2+616 b^3 B\right )\right )+240 \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+3696 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{27720 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{3} \sec \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \sec \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.81, size = 825, normalized size = 2.39 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
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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