Optimal. Leaf size=35 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.007553, antiderivative size = 35, 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} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{(b \sec (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \cos (c+d x) \, dx}{b \sqrt{b \sec (c+d x)}}\\ &=\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{b d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0410732, size = 32, normalized size = 0.91 \[ \frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 41, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d\cos \left ( dx+c \right ) }\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.93042, size = 18, normalized size = 0.51 \begin{align*} \frac{\sin \left (d x + c\right )}{b^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65839, size = 84, normalized size = 2.4 \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^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.5518, size = 36, normalized size = 1.03 \begin{align*} \begin{cases} \frac{\tan{\left (c + d x \right )}}{b^{\frac{3}{2}} d \sec{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \sqrt{\sec{\left (c \right )}}}{\left (b \sec{\left (c \right )}\right )^{\frac{3}{2}}} & \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{\sec \left (d x + c\right )}}{\left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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