Optimal. Leaf size=285 \[ \frac{2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.842625, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3050, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (8 a^2 C+63 A b^2+45 b^2 C\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3050
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int (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 \frac{(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (\frac{1}{2} a (9 A+C)+\frac{1}{2} b (9 A+7 C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{1}{63} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} a^2 (63 A+13 C)+\frac{1}{2} a b (63 A+43 C) \cos (c+d x)+\frac{1}{4} \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{1}{315} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{8} a^3 (63 A+13 C)+\frac{21}{8} b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \cos (c+d x)+\frac{15}{8} a \left (63 A b^2+8 a^2 C+45 b^2 C\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (63 A b^2+8 a^2 C+45 b^2 C\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{1}{945} \left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{16} a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right )+\frac{63}{16} b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (63 A b^2+8 a^2 C+45 b^2 C\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{1}{21} \left (a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (b \left (9 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{2 b \left (9 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)}}{15 d}+\frac{2 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 b \left (24 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (63 A b^2+8 a^2 C+45 b^2 C\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.67249, size = 203, normalized size = 0.71 \[ \frac{\sqrt{\sec (c+d x)} \left (\sin (2 (c+d x)) \left (7 b \left (108 a^2 C+36 A b^2+43 b^2 C\right ) \cos (c+d x)+5 \left (84 a^3 C+252 a A b^2+54 a b^2 C \cos (2 (c+d x))+234 a b^2 C+7 b^3 C \cos (3 (c+d x))\right )\right )+120 a \left (7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+168 b \left (9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.281, size = 718, normalized size = 2.5 \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{\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 ({\left (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}\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{\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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