Optimal. Leaf size=47 \[ \frac {4 \sqrt {a+a \sin (c+d x)}}{a^2 d}-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 a^3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2746, 45}
\begin {gather*} \frac {4 \sqrt {a \sin (c+d x)+a}}{a^2 d}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {a-x}{\sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {2 a}{\sqrt {a+x}}-\sqrt {a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {4 \sqrt {a+a \sin (c+d x)}}{a^2 d}-\frac {2 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 0.68 \begin {gather*} -\frac {2 (-5+\sin (c+d x)) \sqrt {a (1+\sin (c+d x))}}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 29, normalized size = 0.62
method | result | size |
default | \(-\frac {2 \sqrt {a +a \sin \left (d x +c \right )}\, \left (\sin \left (d x +c \right )-5\right )}{3 a^{2} d}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 36, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 6 \, \sqrt {a \sin \left (d x + c\right ) + a} a\right )}}{3 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 28, normalized size = 0.60 \begin {gather*} -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sin \left (d x + c\right ) - 5\right )}}{3 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.72, size = 65, normalized size = 1.38 \begin {gather*} -\frac {4 \, {\left (\sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{3 \, a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \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 \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \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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