Integrand size = 23, antiderivative size = 54 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=-\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}+\frac {2 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{d} \]
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Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3906, 3886, 221, 3892} \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {2 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}-\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d} \]
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Rule 221
Rule 3886
Rule 3892
Rule 3906
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx+\int \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)} \, dx \\ & = -\frac {2 \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 {2 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\left (2 \arcsin \left (\sqrt {\sec (c+d x)}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right )\right ) \cot (c+d x) \sqrt {-\tan ^2(c+d x)}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(181\) vs. \(2(50)=100\).
Time = 1.72 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.37
method | result | size |
default | \(\frac {\left (\arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {2}+\arctan \left (\frac {\cos \left (d x +c \right )-\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )-\arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )\right ) \sqrt {1+\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{\frac {3}{2}} \cos \left (d x +c \right )^{2}}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(182\) |
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (50) = 100\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 4.13 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\sqrt {2} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - \log \left (-\frac {\cos \left (d x + c\right )^{2} + 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right )}{2 \, d} \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {\sec ^{\frac {3}{2}}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )} + 1}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (50) = 100\).
Time = 0.43 (sec) , antiderivative size = 473, normalized size of antiderivative = 8.76 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=-\frac {\sqrt {2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 1\right ) - \sqrt {2} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 1\right ) - \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 2\right ) + \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) - 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 2\right ) - \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} - 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 2\right ) + \log \left (2 \, \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right )^{2} - 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) - 2 \, \sqrt {2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right )\right )\right ) + 2\right )}{2 \, d} \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {\sec \left (d x + c\right ) + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}} \,d x \]
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