Optimal. Leaf size=184 \[ \frac {64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (21 A+15 B+13 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {2 a (21 A+15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac {2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d} \]
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Rubi [A] time = 0.25, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3023, 2751, 2647, 2646} \[ \frac {16 a^2 (21 A+15 B+13 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a (21 A+15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac {2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rule 3023
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {2 \int (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (9 A+7 C)+\frac {1}{2} a (9 B-2 C) \cos (c+d x)\right ) \, dx}{9 a}\\ &=\frac {2 (9 B-2 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {1}{21} (21 A+15 B+13 C) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac {2 a (21 A+15 B+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {1}{105} (8 a (21 A+15 B+13 C)) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (21 A+15 B+13 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (21 A+15 B+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}+\frac {1}{315} \left (32 a^2 (21 A+15 B+13 C)\right ) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (21 A+15 B+13 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (21 A+15 B+13 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 d}+\frac {2 (9 B-2 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{63 d}+\frac {2 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{9 a d}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 114, normalized size = 0.62 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} ((2352 A+3030 B+3116 C) \cos (c+d x)+4 (63 A+180 B+254 C) \cos (2 (c+d x))+7476 A+90 B \cos (3 (c+d x))+6240 B+260 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+5653 C)}{1260 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 124, normalized size = 0.67 \[ \frac {2 \, {\left (35 \, C a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B + 26 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 60 \, B + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (294 \, A + 345 \, B + 292 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (903 \, A + 690 \, B + 584 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.78, size = 336, normalized size = 1.83 \[ \frac {1}{2520} \, \sqrt {2} {\left (\frac {35 \, C a^{2} \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 {45 \, {\left (2 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a^{2} \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 {126 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, C a^{2} \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 {210 \, {\left (10 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 11 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 10 \, C a^{2} \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 {630 \, {\left (12 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 8 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a^{2} \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 {630 \, {\left (8 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, C a^{2} \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 = 0.71, size = 132, normalized size = 0.72 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (140 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90 B -540 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (63 A +315 B +819 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-210 A -420 B -630 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 A +315 B +315 C \right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 230, normalized size = 1.25 \[ \frac {84 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 25 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 150 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 30 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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