Integrand size = 23, antiderivative size = 85 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}-\frac {\text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{d}+\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}} \]
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Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3907, 4108, 3892, 221, 3886} \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac {\text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}} \]
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Rule 221
Rule 3886
Rule 3892
Rule 3907
Rule 4108
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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} \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 {\text {arcsinh}\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*}
Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\left (\arcsin \left (\sqrt {1-\sec (c+d x)}\right )+2 \arcsin \left (\sqrt {\sec (c+d x)}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right )+\sqrt {-((-1+\sec (c+d x)) \sec (c+d x))}\right ) \tan (c+d x)}{d \sqrt {-\tan ^2(c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(77)=154\).
Time = 1.51 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.72
method | result | size |
default | \(\frac {\sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {1+\sec \left (d x +c \right )}\, \left (-2 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{3} \sqrt {2}+2 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )-\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )+\cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )\right )}{2 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(231\) |
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Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (77) = 154\).
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.49 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {2 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) - {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + \frac {4 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 873 vs. \(2 (77) = 154\).
Time = 0.46 (sec) , antiderivative size = 873, normalized size of antiderivative = 10.27 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right ) + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}} \,d x \]
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