Optimal. Leaf size=225 \[ \frac {2 a^2 (33 A+28 C) \sin (c+d x) \cos ^3(c+d x)}{231 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (143 A+112 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{385 d}-\frac {4 a (143 A+112 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{1155 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {2 a C \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{33 d} \]
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Rubi [A] time = 0.64, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3046, 2976, 2981, 2759, 2751, 2646} \[ \frac {2 a^2 (33 A+28 C) \sin (c+d x) \cos ^3(c+d x)}{231 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (143 A+112 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{385 d}-\frac {4 a (143 A+112 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{1155 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {2 a C \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{33 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2976
Rule 2981
Rule 3046
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+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 {3}{2} a C \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 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 {9}{4} a^2 (11 A+8 C)+\frac {3}{4} a^2 (33 A+28 C) \cos (c+d x)\right ) \, dx}{99 a}\\ &=\frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 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}{77} (a (143 A+112 C)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 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}{385} (2 (143 A+112 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (143 A+112 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1155 d}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 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}{165} (a (143 A+112 C)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (143 A+112 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1155 d}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 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 = 0.96, size = 115, normalized size = 0.51 \[ \frac {a \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (2 (5566 A+5789 C) \cos (c+d x)+8 (429 A+581 C) \cos (2 (c+d x))+660 A \cos (3 (c+d x))+21736 A+1645 C \cos (3 (c+d x))+490 C \cos (4 (c+d x))+105 C \cos (5 (c+d x))+18494 C)}{9240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 119, normalized size = 0.53 \[ \frac {2 \, {\left (105 \, C a \cos \left (d x + c\right )^{5} + 245 \, C a \cos \left (d x + c\right )^{4} + 5 \, {\left (33 \, A + 56 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (143 \, A + 112 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \, {\left (143 \, A + 112 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (143 \, A + 112 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 275, normalized size = 1.22 \[ \frac {1}{18480} \, \sqrt {2} {\left (\frac {105 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {385 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {165 \, {\left (4 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {231 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {770 \, {\left (10 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 9 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {2310 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {9240 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 137, normalized size = 0.61 \[ \frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-1680 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6160 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-660 A -9240 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1848 A +7392 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1925 A -3465 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 A +1155 C \right ) \sqrt {2}}{1155 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 170, normalized size = 0.76 \[ \frac {44 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 7 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 429 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 990 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3630 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{18480 \, d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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