Optimal. Leaf size=151 \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.291383, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {4222, 2779, 2984, 12, 2782, 205} \[ \frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 4222
Rule 2779
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{a-2 a \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int -\frac{3 a^2}{2 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx\\ &=-\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}-\frac{\left (2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{\sqrt{a} d}-\frac{2 \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.56287, size = 475, normalized size = 3.15 \[ -\frac{2 \left (\frac{1}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}\right )^{7/2} \cot \left (\frac{c}{2}+\frac{d x}{2}\right ) \csc ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (12 \sin ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right ) \, _3F_2\left (2,2,\frac{7}{2};1,\frac{9}{2};\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}\right )+12 \left (3 \sin ^4\left (\frac{c}{2}+\frac{d x}{2}\right )-7 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )+4\right ) \sin ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}\right )+7 \left (1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3 \sqrt{\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}} \left (8 \sin ^4\left (\frac{c}{2}+\frac{d x}{2}\right )-20 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )+15\right ) \left (\sqrt{\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}} \left (7 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-3\right )+\left (3-6 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \tanh ^{-1}\left (\sqrt{\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )-1}}\right )\right )\right )}{63 d \sqrt{a (\cos (c+d x)+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.44, size = 227, normalized size = 1.5 \begin{align*}{\frac{\sqrt{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,da \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +6\,\cos \left ( dx+c \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +3\, \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{2}-\sqrt{2}\sin \left ( dx+c \right ) \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85062, size = 358, normalized size = 2.37 \begin{align*} -\frac{\frac{3 \, \sqrt{2}{\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right )}{\sqrt{a}} + \frac{2 \, \sqrt{a \cos \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{3 \,{\left (a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right )\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{a \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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