Optimal. Leaf size=243 \[ \frac {2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac {4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{99 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]
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Rubi [A] time = 0.70, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3045, 2976, 2981, 2759, 2751, 2646} \[ \frac {2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac {4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{99 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2976
Rule 2981
Rule 3045
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {2 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (11 A+6 C)+\frac {1}{2} a (11 B+3 C) \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {4 \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {3}{4} a^2 (33 A+22 B+24 C)+\frac {1}{4} a^2 (99 A+110 B+84 C) \cos (c+d x)\right ) \, dx}{99 a}\\ &=\frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {1}{231} (a (429 A+374 B+336 C)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {(2 (429 A+374 B+336 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{1155}\\ &=\frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {1}{495} (a (429 A+374 B+336 C)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 1.31, size = 145, normalized size = 0.60 \[ \frac {a \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} ((33396 A+35156 B+34734 C) \cos (c+d x)+8 (1287 A+1507 B+1743 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+65208 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+4935 C \cos (3 (c+d x))+1470 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+55482 C)}{27720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 137, normalized size = 0.56 \[ \frac {2 \, {\left (315 \, C a \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, B + 21 \, C\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (99 \, A + 187 \, B + 168 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.35, size = 361, normalized size = 1.49 \[ \frac {1}{55440} \, \sqrt {2} {\left (\frac {315 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {385 \, {\left (2 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {495 \, {\left (4 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {693 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 12 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {2310 \, {\left (10 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 10 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 9 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {6930 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 8 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {27720 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 154, normalized size = 0.63 \[ \frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-5040 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3080 B +18480 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1980 A -9900 B -27720 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5544 A +12474 B +22176 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5775 A -8085 B -10395 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B +3465 C \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 252, normalized size = 1.04 \[ \frac {132 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 22 \, {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + 21 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 429 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 990 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3630 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{55440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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