Optimal. Leaf size=32 \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.0066801, 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 = {17, 2637} \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \sec (c+d x)} \int \cos (c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0355135, size = 32, normalized size = 1. \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 41, normalized size = 1.3 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d\cos \left ( dx+c \right ) }\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.05672, size = 18, normalized size = 0.56 \begin{align*} \frac{\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 = 1.40192, size = 76, normalized size = 2.38 \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 )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.9001, size = 36, normalized size = 1.12 \begin{align*} \begin{cases} \frac{\sqrt{b} \tan{\left (c + d x \right )}}{d \sec{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \sqrt{b \sec{\left (c \right )}}}{\sec ^{\frac{3}{2}}{\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{\sqrt{b \sec \left (d x + c\right )}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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