Optimal. Leaf size=32 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0066139, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {18, 2637} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 2637
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \cos (c+d x) \, dx}{\sqrt{b \sec (c+d x)}}\\ &=\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0361462, size = 32, normalized size = 1. \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 41, normalized size = 1.3 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.09274, size = 18, normalized size = 0.56 \begin{align*} \frac{\sin \left (d x + c\right )}{\sqrt{b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7484, size = 81, normalized size = 2.53 \begin{align*} \frac{\sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.9602, size = 36, normalized size = 1.12 \begin{align*} \begin{cases} \frac{\tan{\left (c + d x \right )}}{\sqrt{b} d \sec{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\sqrt{b \sec{\left (c \right )}} \sqrt{\sec{\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{1}{\sqrt{b \sec \left (d x + c\right )} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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