Integrand size = 23, antiderivative size = 105 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {7 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan (c+d x)}{a d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3884, 4086, 3880, 209} \[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=-\frac {7 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {2 \tan (c+d x)}{a d \sqrt {a \sec (c+d x)+a}}+\frac {\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}} \]
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Rule 209
Rule 3880
Rule 3884
Rule 4086
Rubi steps \begin{align*} \text {integral}& = \frac {\tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\sec (c+d x) \left (-\frac {3 a}{2}+2 a \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{2 a^2} \\ & = \frac {\tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan (c+d x)}{a d \sqrt {a+a \sec (c+d x)}}-\frac {7 \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a} \\ & = \frac {\tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan (c+d x)}{a d \sqrt {a+a \sec (c+d x)}}+\frac {7 \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2 a d} \\ & = -\frac {7 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {2 \tan (c+d x)}{a d \sqrt {a+a \sec (c+d x)}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\left (-7 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right ) (1+\sec (c+d x))+2 \sqrt {1-\sec (c+d x)} (5+4 \sec (c+d x))\right ) \tan (c+d x)}{4 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{3/2}} \]
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Time = 0.96 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.38
method | result | size |
default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+7 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right ) \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}-9 \csc \left (d x +c \right )+9 \cot \left (d x +c \right )\right )}{4 d \,a^{2}}\) | \(145\) |
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Time = 0.33 (sec) , antiderivative size = 336, normalized size of antiderivative = 3.20 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [-\frac {7 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (5 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {7 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (5 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]
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\[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 1.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.50 \[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (\frac {\sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {9 \, \sqrt {2}}{a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} + \frac {7 \, \sqrt {2} \log \left ({\left | -\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{4 \, d} \]
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Timed out. \[ \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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