Optimal. Leaf size=138 \[ \frac {64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (7 A+5 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{105 d}+\frac {2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac {2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2751, 2647, 2646} \[ \frac {64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (7 A+5 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{105 d}+\frac {2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac {2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\frac {2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} (7 A+5 B) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac {2 a (7 A+5 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{35} (8 a (7 A+5 B)) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (7 A+5 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (7 A+5 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (7 A+5 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (7 A+5 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 B (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 83, normalized size = 0.60 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} ((392 A+505 B) \cos (c+d x)+6 (7 A+20 B) \cos (2 (c+d x))+1246 A+15 B \cos (3 (c+d x))+1040 B)}{210 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 95, normalized size = 0.69 \[ \frac {2 \, {\left (15 \, B a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (98 \, A + 115 \, B\right )} a^{2} \cos \left (d x + c\right ) + {\left (301 \, A + 230 \, B\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 225, normalized size = 1.63 \[ \frac {1}{420} \, \sqrt {2} {\left (\frac {15 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {21 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {35 \, {\left (10 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 11 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {105 \, {\left (8 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {420 \, {\left (3 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 104, normalized size = 0.75 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-30 B \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 A +105 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-70 A -140 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 A +105 B \right ) \sqrt {2}}{105 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 139, normalized size = 1.01 \[ \frac {14 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 25 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 150 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 5 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a}}{420 \, d} \]
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 \[ \int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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