Optimal. Leaf size=382 \[ \frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac{4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{2 \left (11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6435 d}+\frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac{16 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d} \]
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Rubi [A] time = 1.1538, antiderivative size = 382, 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 = {3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac{4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{2 \left (11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6435 d}+\frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac{16 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d} \]
Antiderivative was successfully verified.
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Rule 3050
Rule 3049
Rule 3033
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{2}{13} \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} b (13 A+11 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{4}{143} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac{1}{4} a^2 (143 A+57 C)+\frac{1}{2} a b (143 A+113 C) \cos (c+d x)+\frac{1}{4} \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{8 \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac{3}{8} a \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac{1}{8} b \left (77 b^2 (13 A+11 C)+3 a^2 (1287 A+961 C)\right ) \cos (c+d x)+\frac{1}{4} a \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac{4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{16 \int \sqrt{\cos (c+d x)} \left (\frac{21}{16} a^2 \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac{117}{4} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac{7}{16} \left (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 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{32 \int \sqrt{\cos (c+d x)} \left (\frac{231}{32} \left (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}{8} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045}\\ &=\frac{2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{1}{77} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{195} \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac{1}{231} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{195 d}+\frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac{4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac{2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac{16 a C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac{2 C \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}\\ \end{align*}
Mathematica [A] time = 3.01578, size = 281, normalized size = 0.74 \[ \frac{24960 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+7392 \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (154 \left (156 a^2 b^2 (36 A+43 C)+936 a^4 C+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 b \left (3744 a \left (11 a^2 C+11 A b^2+16 b^2 C\right ) \cos (2 (c+d x))+77 \left (312 a^2 b C+52 A b^3+89 b^3 C\right ) \cos (3 (c+d x))+312 a \left (44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right )+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right )}{720720 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.531, size = 1017, 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] 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{\cos \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{\cos \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(-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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