Optimal. Leaf size=22 \[ \frac{2 \sqrt{a+b \sin (c+d x)}}{b d} \]
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Rubi [A] time = 0.0353723, antiderivative size = 22, normalized size of antiderivative = 1., 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 2668
Rule 32
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*}
Mathematica [A] time = 0.0134846, size = 22, normalized size = 1. \[ \frac{2 \sqrt{a+b \sin (c+d x)}}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{a+b\sin \left ( dx+c \right ) }}{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964226, size = 27, normalized size = 1.23 \begin{align*} \frac{2 \, \sqrt{b \sin \left (d x + c\right ) + a}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95857, size = 46, normalized size = 2.09 \begin{align*} \frac{2 \, \sqrt{b \sin \left (d x + c\right ) + a}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.02259, size = 54, normalized size = 2.45 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{\sqrt{a}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{\sqrt{a + b \sin{\left (c \right )}}} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{\sqrt{a} d} & \text{for}\: b = 0 \\\frac{2 \sqrt{a + b \sin{\left (c + d x \right )}}}{b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09504, size = 27, normalized size = 1.23 \begin{align*} \frac{2 \, \sqrt{b \sin \left (d x + c\right ) + a}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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