Optimal. Leaf size=59 \[ -\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.03, 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 = {2647, 2646} \[ -\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{3} (4 a) \int \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 89, normalized size = 1.51 \[ -\frac {(a (\sin (c+d x)+1))^{3/2} \left (-9 \sin \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {3}{2} (c+d x)\right )+9 \cos \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {3}{2} (c+d x)\right )\right )}{3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 76, normalized size = 1.29 \[ -\frac {2 \, {\left (a \cos \left (d x + c\right )^{2} + 5 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right ) - 4 \, a\right )} \sin \left (d x + c\right ) + 4 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 101, normalized size = 1.71 \[ -\frac {1}{3} \, \sqrt {2} {\left (\frac {3 \, a \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {a \cos \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {6 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 53, normalized size = 0.90 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right ) \left (\sin \left (d x +c \right )+5\right )}{3 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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