Optimal. Leaf size=22 \[ \frac {2 \sqrt {a+b \sin (c+d x)}}{b d} \]
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Rubi [A] time = 0.04, 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 = {2668, 32} \[ \frac {2 \sqrt {a+b \sin (c+d x)}}{b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+x}} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {2 \sqrt {a+b \sin (c+d x)}}{b d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {2 \sqrt {a+b \sin (c+d x)}}{b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 20, normalized size = 0.91 \[ \frac {2 \, \sqrt {b \sin \left (d x + c\right ) + a}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.79, size = 20, normalized size = 0.91 \[ \frac {2 \, \sqrt {b \sin \left (d x + c\right ) + a}}{b d} \]
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 \[ \frac {2 \sqrt {a +b \sin \left (d x +c \right )}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 20, normalized size = 0.91 \[ \frac {2 \, \sqrt {b \sin \left (d x + c\right ) + a}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.22, size = 20, normalized size = 0.91 \[ \frac {2\,\sqrt {a+b\,\sin \left (c+d\,x\right )}}{b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.14, size = 54, normalized size = 2.45 \[ \begin {cases} \frac {x \cos {\relax (c )}}{\sqrt {a}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{\sqrt {a} d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{\sqrt {a + b \sin {\relax (c )}}} & \text {for}\: d = 0 \\\frac {2 \sqrt {a + b \sin {\left (c + d x \right )}}}{b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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