Optimal. Leaf size=28 \[ \frac {2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2749} \[ \frac {2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2749
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{3/2} \left (-\frac {3 B}{5}+B \cos (c+d x)\right ) \, dx &=\frac {2 B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 45, normalized size = 1.61 \[ \frac {8 a B \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 36, normalized size = 1.29 \[ \frac {2 \, {\left (B a \cos \left (d x + c\right ) + B a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 86, normalized size = 3.07 \[ \frac {1}{10} \, \sqrt {2} {\left (\frac {B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {3 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {2 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 48, normalized size = 1.71 \[ \frac {8 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) B \sqrt {2}}{5 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 92, normalized size = 3.29 \[ \frac {{\left (\sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} - 2 \, {\left (\sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 9 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int -\left (\frac {3\,B}{5}-B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B \left (\int \left (- 3 a \sqrt {a \cos {\left (c + d x \right )} + a}\right )\, dx + \int 2 a \sqrt {a \cos {\left (c + d x \right )} + a} \cos {\left (c + d x \right )}\, dx + \int 5 a \sqrt {a \cos {\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}\, dx\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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