Optimal. Leaf size=36 \[ \frac{\sqrt{\sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0074694, antiderivative size = 36, 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, 3770} \[ \frac{\sqrt{\sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \sec (c+d x) \, dx}{b \sqrt{b \sec (c+d x)}}\\ &=\frac{\tanh ^{-1}(\sin (c+d x)) \sqrt{\sec (c+d x)}}{b d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0262157, size = 33, normalized size = 0.92 \[ \frac{\sec ^{\frac{3}{2}}(c+d x) \tanh ^{-1}(\sin (c+d x))}{d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 52, normalized size = 1.4 \begin{align*} -2\,{\frac{\cos \left ( dx+c \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{5/2}}{d}{\it Artanh} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-3/2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.05257, size = 88, normalized size = 2.44 \begin{align*} \frac{\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}{2 \, b^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88568, size = 301, normalized size = 8.36 \begin{align*} \left [\frac{\log \left (-\frac{b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b}{\cos \left (d x + c\right )^{2}}\right )}{2 \, b^{\frac{3}{2}} d}, -\frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b}\right )}{b^{2} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sec \left (d x + c\right )^{\frac{5}{2}}}{\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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