Optimal. Leaf size=77 \[ \frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.108971, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {3805, 3804} \[ \frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2}{3} \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{4 a \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.154257, size = 49, normalized size = 0.64 \[ \frac{2 (\cos (c+d x)+2) \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)}}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.21, size = 68, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2\,\cos \left ( dx+c \right ) -4 \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\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 = 2.54467, size = 153, normalized size = 1.99 \begin{align*} \frac{\sqrt{2}{\left (3 \, \cos \left (\frac{2}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) - 3 \, \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) \sin \left (\frac{2}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) + 2 \, \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right )\right )} \sqrt{a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65563, size = 181, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right ) + d\right )} \sqrt{\cos \left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}{\sec ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \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{a \sec \left (d x + c\right ) + a}}{\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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