Optimal. Leaf size=116 \[ \frac {64 a^3 \sin (c+d x)}{21 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{21 d}+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2751, 2647, 2646} \[ \frac {64 a^3 \sin (c+d x)}{21 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{21 d}+\frac {2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx &=\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {5}{7} \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} (8 a) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{21} \left (32 a^2\right ) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {64 a^3 \sin (c+d x)}{21 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 84, normalized size = 0.72 \[ \frac {a^2 \left (315 \sin \left (\frac {1}{2} (c+d x)\right )+77 \sin \left (\frac {3}{2} (c+d x)\right )+3 \left (7 \sin \left (\frac {5}{2} (c+d x)\right )+\sin \left (\frac {7}{2} (c+d x)\right )\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{84 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 75, normalized size = 0.65 \[ \frac {2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) + 46 \, a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{21 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 117, normalized size = 1.01 \[ \frac {1}{84} \, \sqrt {2} {\left (\frac {3 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {21 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {77 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {315 \, a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 86, normalized size = 0.74 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right ) \sqrt {2}}{21 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 77, normalized size = 0.66 \[ \frac {{\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{84 \, d} \]
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 \[ \int \cos \left (c+d\,x\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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