Optimal. Leaf size=401 \[ \frac{2 \sin (c+d x) \left (242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{15 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 1.06061, antiderivative size = 401, 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 = {4221, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \sin (c+d x) \left (242 a^2 b B+24 a^3 C+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{15 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac{1}{2} a (11 A+3 C)+\frac{1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac{1}{2} (11 b B+6 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac{3}{4} a (33 a A+11 b B+15 a C)+\frac{1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+150 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{693} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{21}{8} a^2 (33 a A+11 b B+15 a C)+\frac{9}{8} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \cos (c+d x)+\frac{7}{8} \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{231}{16} \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right )+\frac{45}{16} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465}\\ &=\frac{2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{15} \left (\left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{77} \left (\left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 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 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{231} \left (\left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 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 (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 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 (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 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 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.47653, size = 304, normalized size = 0.76 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (240 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (33 a^2 b (7 A+5 C)+77 a^3 B+165 a b^2 B+5 b^3 (11 A+9 C)\right )+3696 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )+\frac{\sin (2 (c+d x)) \left (154 \cos (c+d x) \left (108 a^2 b B+36 a^3 C+3 a b^2 (36 A+43 C)+43 b^3 B\right )+5 \left (36 b \cos (2 (c+d x)) \left (33 a^2 C+33 a b B+11 A b^2+16 b^2 C\right )+396 a^2 b (14 A+13 C)+1848 a^3 B+154 b^2 (3 a C+b B) \cos (3 (c+d x))+5148 a b^2 B+3 b^3 (572 A+531 C)+63 b^3 C \cos (4 (c+d x))\right )\right )}{\sqrt{\cos (c+d x)}}\right )}{27720 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.485, size = 1082, normalized size = 2.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \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 (\frac{C b^{3} \cos \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \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 )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\sqrt{\sec \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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