Optimal. Leaf size=423 \[ \frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 (202 A b+350 b C)+63 a^3 B+413 a b^2 B+192 A b^3\right )}{105 d}-\frac{2 b^2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right )}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right )}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d} \]
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Rubi [A] time = 1.46859, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3047, 3031, 3023, 2748, 2641, 2639} \[ \frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left (a^2 (202 A b+350 b C)+63 a^3 B+413 a b^2 B+192 A b^3\right )}{105 d}-\frac{2 b^2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \cos (c+d x))^2}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right )}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right )}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d} \]
Antiderivative was successfully verified.
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Rule 4221
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+B \cos (c+d x)+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+B \cos (c+d x)+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 (\frac{1}{2} (8 A b+7 a B)+\frac{1}{2} (5 a A+7 b B+7 a 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{2 (8 A b+7 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+77 a b B+5 a^2 (5 A+7 C)\right )+\frac{1}{4} \left (34 a A b+21 a^2 B+35 b^2 B+70 a b C\right ) \cos (c+d x)-\frac{1}{4} b (39 A b+21 a B-35 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 \left (48 A b^2+77 a b B+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{2 (8 A b+7 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}{8} \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right )+\frac{1}{8} \left (77 a^2 b B+105 b^3 B+5 a^3 (5 A+7 C)+3 a b^2 (11 A+105 C)\right ) \cos (c+d x)-\frac{3}{8} b \left (98 a b B+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{2 a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+77 a b B+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{2 (8 A b+7 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^4-140 a^3 b B-518 a b^3 B-5 a^4 (5 A+7 C)-5 a^2 b^2 (47 A+133 C)\right )+\frac{21}{16} \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \cos (c+d x)+\frac{3}{16} b^2 \left (98 a b B+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 (98 a b B+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{2 a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+77 a b B+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{2 (8 A b+7 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 (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac{63}{32} \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b^2 \left (98 a b B+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{2 a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+77 a b B+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{2 (8 A b+7 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 (\left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (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 (28 a^3 b B+84 a b^3 B+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{2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (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 (28 a^3 b B+84 a b^3 B+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 (98 a b B+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{2 a \left (192 A b^3+63 a^3 B+413 a b^2 B+a^2 (202 A b+350 b C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d}+\frac{2 \left (48 A b^2+77 a b B+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{2 (8 A b+7 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 = 4.43897, size = 294, normalized size = 0.7 \[ \frac{\sqrt{\sec (c+d x)} \left (42 a \sin (c+d x) \left (4 a^2 b (3 A+5 C)+3 a^3 B+30 a b^2 B+20 A b^3\right )+10 a^2 \tan (c+d x) \left (a^2 (5 A+7 C)+28 a b B+42 A b^2\right )+10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)\right )-42 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right )+42 a^3 (a B+4 A b) \tan (c+d x) \sec (c+d x)+30 a^4 A \tan (c+d x) \sec ^2(c+d x)+35 b^4 C \sin (2 (c+d x))\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.289, size = 1624, normalized size = 3.8 \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} + B \cos \left (d x + c\right ) + 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} +{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\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} + B \cos \left (d x + c\right ) + 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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