Optimal. Leaf size=22 \[ -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 32} \[ -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {2}{a d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ -\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 33, normalized size = 1.50 \[ -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.88, size = 20, normalized size = 0.91 \[ -\frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 0.95 \[ -\frac {2}{a d \sqrt {a +a \sin \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 20, normalized size = 0.91 \[ -\frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.88, size = 50, normalized size = 2.27 \[ -\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (c+d\,x\right )+1\right )}{a^2\,d\,\left (2\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.46, size = 56, normalized size = 2.55 \[ \begin {cases} \text {NaN} & \text {for}\: \left (c = \frac {3 \pi }{2} \vee c = - d x + \frac {3 \pi }{2}\right ) \wedge \left (c = - d x + \frac {3 \pi }{2} \vee d = 0\right ) \\\frac {x \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2}{a d \sqrt {a \sin {\left (c + d x \right )} + a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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