Optimal. Leaf size=59 \[ \frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.012328, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {17, 2637} \[ \frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin{align*} \int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx &=\frac{\sqrt{b \cos (c+d x)} \int (A+B \cos (c+d x)) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{\left (B \sqrt{b \cos (c+d x)}\right ) \int \cos (c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0469873, size = 42, normalized size = 0.71 \[ \frac{\sqrt{b \cos (c+d x)} (A (c+d x)+B \sin (c+d x))}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.455, size = 39, normalized size = 0.7 \begin{align*}{\frac{A \left ( dx+c \right ) +B\sin \left ( dx+c \right ) }{d}\sqrt{b\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.99071, size = 54, normalized size = 0.92 \begin{align*} \frac{2 \, A \sqrt{b} \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + B \sqrt{b} \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61787, size = 508, normalized size = 8.61 \begin{align*} \left [\frac{A \sqrt{-b} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, \sqrt{b \cos \left (d x + c\right )} B \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )}, \frac{A \sqrt{b} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{b} \cos \left (d x + c\right )^{\frac{3}{2}}}\right ) \cos \left (d x + c\right ) + \sqrt{b \cos \left (d x + c\right )} B \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.2536, size = 46, normalized size = 0.78 \begin{align*} \begin{cases} A \sqrt{b} x + \frac{B \sqrt{b} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\\frac{x \sqrt{b \cos{\left (c \right )}} \left (A + B \cos{\left (c \right )}\right )}{\sqrt{\cos{\left (c \right )}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \cos \left (d x + c\right ) + A\right )} \sqrt{b \cos \left (d x + c\right )}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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