Optimal. Leaf size=356 \[ \frac{4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}-\frac{2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{8 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d} \]
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Rubi [A] time = 1.3014, antiderivative size = 356, 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, 3048, 3047, 3031, 3023, 2748, 2641, 2639} \[ \frac{4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}-\frac{2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{8 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3048
Rule 3047
Rule 3031
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 ) \sec ^{\frac{9}{2}}(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 )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{7} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^3 \left (4 A b+\frac{1}{2} a (5 A+7 C) \cos (c+d x)-\frac{1}{2} b (3 A-7 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{16 A b (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{35} \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} \left (48 A b^2+5 a^2 (5 A+7 C)\right )+\frac{1}{2} a b (17 A+35 C) \cos (c+d x)-\frac{1}{4} b^2 (39 A-35 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{16 A b (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{105} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} b \left (96 A b^2+a^2 (101 A+175 C)\right )+\frac{1}{8} a \left (5 a^2 (5 A+7 C)+3 b^2 (11 A+105 C)\right ) \cos (c+d x)-\frac{3}{8} b \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{16 A b (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{105} \left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{16} \left (-192 A b^4-5 a^4 (5 A+7 C)-5 a^2 b^2 (47 A+133 C)\right )+\frac{21}{4} a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \cos (c+d x)+\frac{3}{16} b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{16 A b (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{315} \left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{15}{32} \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac{63}{8} a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{16 A b (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac{1}{5} \left (4 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (\left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 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 (5 b^2 (A-C)+a^2 (3 A+5 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 \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 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^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{16 A b (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+b \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 3.27073, size = 296, normalized size = 0.83 \[ \frac{2 \sec ^{\frac{7}{2}}(c+d x) \left (5 \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\right ) \cos ^{\frac{7}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-84 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \cos ^{\frac{7}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+210 a^2 A b^2 \sin (c+d x) \cos ^2(c+d x)+42 a^3 A b \sin (2 (c+d x))+252 a^3 A b \sin (c+d x) \cos ^3(c+d x)+15 a^4 A \sin (c+d x)+25 a^4 A \sin (c+d x) \cos ^2(c+d x)+420 a^3 b C \sin (c+d x) \cos ^3(c+d x)+35 a^4 C \sin (c+d x) \cos ^2(c+d x)+420 a A b^3 \sin (c+d x) \cos ^3(c+d x)+35 b^4 C \sin (c+d x) \cos ^4(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.316, size = 1531, normalized size = 4.3 \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] 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} \sec \left (d x + c\right )^{\frac{9}{2}}\,{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 )} \sec \left (d x + c\right )^{\frac{9}{2}}, 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} \sec \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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