Integrand size = 32, antiderivative size = 184 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {7 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{5/2} d}-\frac {79 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {23 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}} \]
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Time = 0.58 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4105, 4107, 4005, 3859, 209, 3880} \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {7 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{5/2} d}-\frac {79 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}+\frac {23 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}} \]
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Rule 209
Rule 3859
Rule 3880
Rule 4005
Rule 4105
Rule 4107
Rubi steps \begin{align*} \text {integral}& = -\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}+\frac {\int \frac {\cos (c+d x) (6 a A+5 a A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos (c+d x) \left (23 a^2 A+\frac {33}{2} a^2 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{8 a^4} \\ & = -\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {23 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {-28 a^3 A-\frac {23}{2} a^3 A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{8 a^5} \\ & = -\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {23 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(7 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{2 a^3}+\frac {(79 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{16 a^2} \\ & = -\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {23 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(7 A) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^2 d}-\frac {(79 A) \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 )}{8 a^2 d} \\ & = \frac {7 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{5/2} d}-\frac {79 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {23 A \sin (c+d x)}{8 a^2 d \sqrt {a-a \sec (c+d x)}} \\ \end{align*}
Time = 1.75 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.03 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {A \left (448 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \sin ^5\left (\frac {1}{2} (c+d x)\right )-316 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \sin ^5\left (\frac {1}{2} (c+d x)\right )+\sqrt {1+\sec (c+d x)} (8 \sin (c+d x)+(-35+23 \sec (c+d x)) \tan (c+d x))\right )}{8 a^2 d (-1+\sec (c+d x))^2 \sqrt {1+\sec (c+d x)} \sqrt {a-a \sec (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(155)=310\).
Time = 31.65 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.04
method | result | size |
default | \(-\frac {A \sqrt {2}\, \left (8 \cos \left (d x +c \right )^{3} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+56 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {2}\, \cos \left (d x +c \right )^{2}-27 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-112 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-12 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \cos \left (d x +c \right )+79 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{2}+56 \sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+23 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-158 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+79 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )\right ) \csc \left (d x +c \right )}{16 a^{2} d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )-1\right ) \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}}\) | \(375\) |
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Time = 0.32 (sec) , antiderivative size = 612, normalized size of antiderivative = 3.33 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\left [-\frac {79 \, \sqrt {2} {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} + {\left (3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 112 \, {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 4 \, {\left (8 \, A \cos \left (d x + c\right )^{4} - 27 \, A \cos \left (d x + c\right )^{3} - 12 \, A \cos \left (d x + c\right )^{2} + 23 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{32 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}, \frac {79 \, \sqrt {2} {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\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 ) \sin \left (d x + c\right ) - 112 \, {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {a} \arctan \left (\frac {\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 ) \sin \left (d x + c\right ) - 2 \, {\left (8 \, A \cos \left (d x + c\right )^{4} - 27 \, A \cos \left (d x + c\right )^{3} - 12 \, A \cos \left (d x + c\right )^{2} + 23 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=A \left (\int \frac {\cos {\left (c + d x \right )}}{a^{2} \sqrt {- a \sec {\left (c + d x \right )} + a} \sec ^{2}{\left (c + d x \right )} - 2 a^{2} \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a^{2} \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx + \int \frac {\cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{a^{2} \sqrt {- a \sec {\left (c + d x \right )} + a} \sec ^{2}{\left (c + d x \right )} - 2 a^{2} \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a^{2} \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx\right ) \]
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\[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 1.24 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.99 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {79 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {112 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {16 \, \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )} a^{2}} - \frac {\sqrt {2} {\left (17 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A + 15 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a\right )}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{16 \, d} \]
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Timed out. \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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