Optimal. Leaf size=369 \[ \frac{4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{8 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 1.24613, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{8 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}} \]
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))^4 \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))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^3 \left (\frac{1}{2} a (11 A+C)+\frac{1}{2} b (11 A+9 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (\frac{1}{4} a^2 (99 A+17 C)+\frac{1}{2} a b (99 A+73 C) \cos (c+d x)+\frac{3}{4} \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{693} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{8} a \left (9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{1}{8} b \left (45 b^2 (11 A+9 C)+a^2 (2079 A+1381 C)\right ) \cos (c+d x)+\frac{1}{4} a \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{16} a^2 \left (9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac{231}{4} a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \cos (c+d x)+\frac{15}{16} \left (64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{3465}\\ &=\frac{4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{32} \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right )+\frac{693}{8} a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{10395}\\ &=\frac{4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}+\frac{1}{15} \left (4 a b \left (3 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}{231} \left (\left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (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{8 a b \left (3 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 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (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{4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.75, size = 265, normalized size = 0.72 \[ \frac{\sqrt{\sec (c+d x)} \left (2 \sin (2 (c+d x)) \left (616 a b \left (36 a^2 C+36 A b^2+43 b^2 C\right ) \cos (c+d x)+5 \left (36 \left (66 a^2 b^2 C+11 A b^4+16 b^4 C\right ) \cos (2 (c+d x))+792 a^2 b^2 (14 A+13 C)+1848 a^4 C+616 a b^3 C \cos (3 (c+d x))+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right )\right )+480 \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+29568 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{55440 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.287, size = 924, 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 )}^{4} \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^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} +{\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 6 \, A a^{2} 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 )}^{4} \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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