Integrand size = 23, antiderivative size = 134 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=-\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}}-\frac {2 \sin (c+d x)}{15 d \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {26 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d \sqrt {1+\sec (c+d x)}} \]
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Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3908, 4107, 4098, 3892, 221} \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=-\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {\sec (c+d x)+1}}+\frac {26 \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d \sqrt {\sec (c+d x)+1}}-\frac {2 \sin (c+d x)}{15 d \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x)+1}} \]
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Rule 221
Rule 3892
Rule 3908
Rule 4098
Rule 4107
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}}-\frac {1}{5} \int \frac {1-4 \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx \\ & = \frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}}-\frac {2 \sin (c+d x)}{15 d \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\frac {2}{15} \int \frac {-\frac {13}{2}+\sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \, dx \\ & = \frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}}-\frac {2 \sin (c+d x)}{15 d \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {26 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d \sqrt {1+\sec (c+d x)}}-\int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx \\ & = \frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}}-\frac {2 \sin (c+d x)}{15 d \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {26 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d \sqrt {1+\sec (c+d x)}}+\frac {\sqrt {2} \text {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} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {1+\sec (c+d x)}}-\frac {2 \sin (c+d x)}{15 d \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {26 \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d \sqrt {1+\sec (c+d x)}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=\frac {\left (15 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right ) \sec ^{\frac {5}{2}}(c+d x)+2 \sqrt {1-\sec (c+d x)} \left (3-\sec (c+d x)+13 \sec ^2(c+d x)\right )\right ) \sin (c+d x)}{15 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {-\tan ^2(c+d x)}} \]
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Time = 1.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.41
method | result | size |
default | \(-\frac {\sqrt {1+\sec \left (d x +c \right )}\, \left (15 \sqrt {2}\, \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \sec \left (d x +c \right )+15 \sqrt {2}\, \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \sec \left (d x +c \right )^{2}-6 \sin \left (d x +c \right )+2 \tan \left (d x +c \right )-26 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{15 d \left (\cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {5}{2}}}\) | \(189\) |
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Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=\frac {15 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + \frac {4 \, {\left (3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 13 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{30 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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\[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {\sec {\left (c + d x \right )} + 1} \sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (114) = 228\).
Time = 0.52 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.64 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} {\left (60 \, \cos \left (\frac {4}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 \, \cos \left (\frac {2}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 60 \, \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \sin \left (\frac {4}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 5 \, \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \sin \left (\frac {2}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) - 30 \, \log \left (\cos \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )^{2} + \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 1\right ) + 30 \, \log \left (\cos \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )^{2} + \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 1\right ) + 6 \, \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 \, \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 60 \, \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )\right )}}{60 \, d} \]
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\[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {\sec \left (d x + c\right ) + 1} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {1+\sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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