Optimal. Leaf size=59 \[ \frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2726, 2725}
\begin {gather*} \frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2726
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{3/2} \, dx &=\frac {2 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} (4 a) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 55, normalized size = 0.93 \begin {gather*} \frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (9 \sin \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {3}{2} (c+d x)\right )\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 58, normalized size = 0.98
method | result | size |
default | \(\frac {4 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sqrt {2}}{3 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 38, normalized size = 0.64 \begin {gather*} \frac {{\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 )} \sqrt {a}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 44, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (a \cos \left (d x + c\right ) + 5 \, a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \end {gather*}
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 \begin {gather*} \int \left (a \cos {\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 55, normalized size = 0.93 \begin {gather*} \frac {\sqrt {2} {\left (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 ) + 9 \, 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 )\right )} \sqrt {a}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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