Optimal. Leaf size=517 \[ \frac{2 \sin (c+d x) \left (11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left (2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 1.52742, antiderivative size = 517, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4221, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \sin (c+d x) \left (11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left (2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
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))^4 \left (A+B \cos (c+d x)+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))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{13} \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))^3 \left (\frac{1}{2} a (13 A+3 C)+\frac{1}{2} (13 A b+13 a B+11 b C) \cos (c+d x)+\frac{1}{2} (13 b B+8 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{143} \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))^2 \left (\frac{1}{4} a (143 a A+39 b B+57 a C)+\frac{1}{4} \left (286 a A b+143 a^2 B+117 b^2 B+226 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (8 \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{3}{8} a \left (338 a b B+11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac{1}{8} \left (1287 a^3 B+2951 a b^2 B+77 b^3 (13 A+11 C)+3 a^2 b (1287 A+961 C)\right ) \cos (c+d x)+\frac{1}{8} \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 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{21}{16} a^2 \left (338 a b B+11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac{117}{16} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \cos (c+d x)+\frac{7}{16} \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^2(c+d x)\right ) \, dx}{9009}\\ &=\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \left (\frac{231}{32} \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right )+\frac{585}{32} \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045}\\ &=\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{1}{77} \left (\left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (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{1}{195} \left (\left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{195 d}+\frac{2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{1}{231} \left (\left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (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 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{195 d}+\frac{2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (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 (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.43192, size = 400, normalized size = 0.77 \[ \frac{\sqrt{\sec (c+d x)} \left (\sin (2 (c+d x)) \left (154 \cos (c+d x) \left (156 a^2 b^2 (36 A+43 C)+3744 a^3 b B+936 a^4 C+4472 a b^3 B+b^4 (1118 A+1171 C)\right )+5 \left (77 b^2 \cos (3 (c+d x)) \left (312 a^2 C+208 a b B+52 A b^2+89 b^2 C\right )+1872 b \cos (2 (c+d x)) \left (33 a^2 b B+22 a^3 C+2 a b^2 (11 A+16 C)+8 b^3 B\right )+78 \left (176 a^3 b (14 A+13 C)+3432 a^2 b^2 B+616 a^4 B+4 a b^3 (572 A+531 C)+531 b^4 B\right )+1638 b^3 (4 a C+b B) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right )+6240 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )+7392 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )\right )}{720720 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.67, size = 1407, normalized size = 2.7 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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 )}{\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} + B \cos \left (d x + c\right ) + 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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