Optimal. Leaf size=85 \[ \frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac{\sinh ^{-1}\left (\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right )}{d} \]
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Rubi [A] time = 0.185332, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3822, 4023, 3807, 215, 3801} \[ \frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac{\sinh ^{-1}\left (\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3822
Rule 4023
Rule 3807
Rule 215
Rule 3801
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx &=\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}+\frac{1}{2} \int \frac{(1-\sec (c+d x)) \sqrt{\sec (c+d x)}}{\sqrt{1+\sec (c+d x)}} \, dx\\ &=\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}-\frac{1}{2} \int \sqrt{\sec (c+d x)} \sqrt{1+\sec (c+d x)} \, dx+\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\sec (c+d x)}} \, dx\\ &=\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{1+\sec (c+d x)}}\right )}{d}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,-\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}\\ &=\frac{\sqrt{2} \sinh ^{-1}\left (\frac{\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}-\frac{\sinh ^{-1}\left (\frac{\tan (c+d x)}{\sqrt{1+\sec (c+d x)}}\right )}{d}+\frac{\sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{1+\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.285565, size = 111, normalized size = 1.31 \[ \frac{\tan (c+d x) \left (\sqrt{-(\sec (c+d x)-1) \sec (c+d x)}+\sin ^{-1}\left (\sqrt{1-\sec (c+d x)}\right )+2 \sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right )\right )}{d \sqrt{-\tan ^2(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.201, size = 218, normalized size = 2.6 \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{\cos \left ( dx+c \right ) +1}{\cos \left ( dx+c \right ) }}} \left ( \arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) }{4}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) \sqrt{2}\cos \left ( dx+c \right ) -\arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) \right ) }{4}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) \sqrt{2}\cos \left ( dx+c \right ) -4\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \cos \left ( dx+c \right ) -2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ){\frac{1}{\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.217, size = 1179, normalized size = 13.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12773, size = 826, normalized size = 9.72 \begin{align*} \frac{2 \,{\left (\sqrt{2} \cos \left (d x + c\right ) + \sqrt{2}\right )} \log \left (\frac{2 \, \sqrt{2} \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) +{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} + 2 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) -{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} - 2 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + \frac{4 \, \sqrt{\frac{\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{4 \,{\left (d \cos \left (d x + c\right ) + 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}}}{\sqrt{\sec \left (d x + c\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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