Optimal. Leaf size=335 \[ \frac{2 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{33 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.927343, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {4221, 3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{33 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3050
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+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+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} b (11 A+9 C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 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{9}{4} a^2 (11 A+5 C)+\frac{3}{2} a b (33 A+25 C) \cos (c+d x)+\frac{3}{4} \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 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{63}{8} a^3 (11 A+5 C)+\frac{9}{8} b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac{21}{8} a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 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{693}{16} a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right )+\frac{45}{16} b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465}\\ &=\frac{2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 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}{5} \left (a \left (a^2 (5 A+3 C)+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 (b \left (33 a^2 (7 A+5 C)+5 b^2 (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 a \left (a^2 (5 A+3 C)+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)}}{5 d}+\frac{2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 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 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{231} \left (b \left (33 a^2 (7 A+5 C)+5 b^2 (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 a \left (a^2 (5 A+3 C)+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)}}{5 d}+\frac{2 b \left (33 a^2 (7 A+5 C)+5 b^2 (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 (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 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 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.40187, size = 236, normalized size = 0.7 \[ \frac{\sqrt{\sec (c+d x)} \left (\sin (2 (c+d x)) \left (154 a \left (12 a^2 C+36 A b^2+43 b^2 C\right ) \cos (c+d x)+5 b \left (12 \left (33 a^2 C+11 A b^2+16 b^2 C\right ) \cos (2 (c+d x))+1848 a^2 A+1716 a^2 C+154 a b C \cos (3 (c+d x))+572 A b^2+21 b^2 C \cos (4 (c+d x))+531 b^2 C\right )\right )+80 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3696 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{9240 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.281, size = 793, normalized size = 2.4 \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} + 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} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} +{\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}}{\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} + 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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