Optimal. Leaf size=22 \[ \frac {2 \sqrt {a \sin (c+d x)+a}}{a d} \]
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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} \[ \frac {2 \sqrt {a \sin (c+d x)+a}}{a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2667
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {2 \sqrt {a+a \sin (c+d x)}}{a d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ \frac {2 \sqrt {a \sin (c+d x)+a}}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, 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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giac [A] time = 0.84, 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 = 0.03, size = 21, normalized size = 0.95 \[ \frac {2 \sqrt {a +a \sin \left (d x +c \right )}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, 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 = 4.82, size = 20, normalized size = 0.91 \[ \frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 32, normalized size = 1.45 \[ \begin {cases} \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\sqrt {a \sin {\relax (c )} + a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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