Optimal. Leaf size=361 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (7 A+9 C)+7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{693 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (33 a^2 b (7 A+9 C)+77 a^3 B+242 a b^2 B+24 A b^3\right )}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac{2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d} \]
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Rubi [A] time = 0.955049, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4112, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (7 A+9 C)+7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{693 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (33 a^2 b (7 A+9 C)+77 a^3 B+242 a b^2 B+24 A b^3\right )}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )}{231 d}+\frac{2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d} \]
Antiderivative was successfully verified.
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Rule 4112
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{2}{11} \int \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \left (\frac{1}{2} b (3 A+11 C)+\frac{1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac{1}{2} (6 A b+11 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 (6 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{4}{99} \int \sqrt{\cos (c+d x)} (b+a \cos (c+d x)) \left (\frac{3}{4} b (15 A b+11 a B+33 b C)+\frac{1}{4} \left (150 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{8}{693} \int \sqrt{\cos (c+d x)} \left (\frac{21}{8} b^2 (15 A b+11 a B+33 b C)+\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 ) \cos (c+d x)+\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 ) \cos ^2(c+d x)\right ) \, dx\\ &=\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 ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac{16 \int \sqrt{\cos (c+d x)} \left (\frac{231}{16} \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 )+\frac{45}{16} \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 ) \cos (c+d x)\right ) \, dx}{3465}\\ &=\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 ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\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 \sqrt{\cos (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 \cos ^{\frac{3}{2}}(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 ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{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)} \sin (c+d x)}{231 d}+\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 ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\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 \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 ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{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 ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 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)} \sin (c+d x)}{231 d}+\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 ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac{2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 2.02615, size = 286, normalized size = 0.79 \[ \frac{10 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right )+154 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (7 A+9 C)+7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)\right )+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left (154 \cos (c+d x) \left (3 a^2 b (43 A+36 C)+43 a^3 B+108 a b^2 B+36 A b^3\right )+5 \left (36 a \cos (2 (c+d x)) \left (a^2 (16 A+11 C)+33 a b B+33 A b^2\right )+3 \left (21 a^3 A \cos (4 (c+d x))+a^3 (531 A+572 C)+1716 a^2 b B+132 a b^2 (13 A+14 C)+616 b^3 B\right )+154 a^2 (a B+3 A b) \cos (3 (c+d x))\right )\right )}{1155 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.615, size = 1082, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{4} + A a^{3} \cos \left (d x + c\right )^{5} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{5} \sec \left (d x + c\right )\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \cos \left (d x + c\right )^{\frac{11}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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