Optimal. Leaf size=477 \[ \frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d}+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d} \]
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Rubi [A] time = 1.32, antiderivative size = 477, normalized size of antiderivative = 1.00, 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 = {3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+468 a^3 b B+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{195 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{1287 d}+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (2171 a^2 b B+192 a^3 C+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{9009 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (11 a^2 b^2 (637 A+491 C)+3458 a^3 b B+192 a^4 C+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{6435 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (44 a^3 b (7 A+5 C)+330 a^2 b^2 B+77 a^4 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )}{231 d}+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rubi steps
\begin {align*} \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 \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} (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) \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 (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 ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 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 (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 ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 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 (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 \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 ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 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 ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 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 (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 \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 ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 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 ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 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 (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 ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \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 ) \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 ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{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)} \sin (c+d x)}{231 d}+\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 ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 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 ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 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 (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 ) \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 ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{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 ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 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)} \sin (c+d x)}{231 d}+\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 ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 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 ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 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*}
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Mathematica [A] time = 3.50, size = 381, normalized size = 0.80 \[ \frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (154 \cos (c+d x) \left (936 a^4 C+3744 a^3 b B+156 a^2 b^2 (36 A+43 C)+4472 a b^3 B+b^4 (1118 A+1171 C)\right )+5 \left (77 b^2 \cos (3 (c+d x)) \left (312 a^2 C+208 a b B+52 A b^2+89 b^2 C\right )+1872 b \cos (2 (c+d x)) \left (22 a^3 C+33 a^2 b B+2 a b^2 (11 A+16 C)+8 b^3 B\right )+78 \left (616 a^4 B+176 a^3 b (14 A+13 C)+3432 a^2 b^2 B+4 a b^3 (572 A+531 C)+531 b^4 B\right )+1638 b^3 (4 a C+b B) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right )+48 \left (65 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right )+77 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )\right )}{360360 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.70, size = 1407, normalized size = 2.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 903, normalized size = 1.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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