Integrand size = 32, antiderivative size = 146 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {5 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {7 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} a^{3/2} d}-\frac {A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {2 A \sin (c+d x)}{a d \sqrt {a-a \sec (c+d x)}} \]
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Time = 0.43 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4105, 4107, 4005, 3859, 209, 3880} \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {5 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {7 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} a^{3/2} d}+\frac {2 A \sin (c+d x)}{a d \sqrt {a-a \sec (c+d x)}}-\frac {A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}} \]
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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)}{d (a-a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos (c+d x) (4 a A+3 a A \sec (c+d x))}{\sqrt {a-a \sec (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {2 A \sin (c+d x)}{a d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {-5 a^2 A-2 a^2 A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{2 a^3} \\ & = -\frac {A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {2 A \sin (c+d x)}{a d \sqrt {a-a \sec (c+d x)}}+\frac {(5 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{2 a^2}+\frac {(7 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{2 a} \\ & = -\frac {A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {2 A \sin (c+d x)}{a d \sqrt {a-a \sec (c+d x)}}+\frac {(5 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 d}-\frac {(7 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 )}{a d} \\ & = \frac {5 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac {7 A \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{\sqrt {2} a^{3/2} d}-\frac {A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac {2 A \sin (c+d x)}{a d \sqrt {a-a \sec (c+d x)}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {A \left (10 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right ) (-1+\sec (c+d x))-7 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right ) (-1+\sec (c+d x))-2 (-2+\cos (c+d x)) \sqrt {1+\sec (c+d x)}\right ) \tan (c+d x)}{2 a d (-1+\sec (c+d x)) \sqrt {1+\sec (c+d x)} \sqrt {a-a \sec (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(260\) vs. \(2(127)=254\).
Time = 29.31 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.79
method | result | size |
default | \(-\frac {A \sqrt {2}\, \left (\cos \left (d x +c \right )^{2} \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+5 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \cos \left (d x +c \right )+7 \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-5 \sqrt {2}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )-2 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-7 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )\right ) \csc \left (d x +c \right )}{2 a d \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-a \left (\sec \left (d x +c \right )-1\right )}}\) | \(261\) |
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Time = 0.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 3.60 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\left [-\frac {7 \, \sqrt {2} {\left (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 ) + 10 \, {\left (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 (A \cos \left (d x + c\right )^{3} - A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{4 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {7 \, \sqrt {2} {\left (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 ) - 10 \, {\left (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 (A \cos \left (d x + c\right )^{3} - A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{2 \, {\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=A \left (\int \frac {\cos {\left (c + d x \right )}}{- a \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx + \int \frac {\cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{- a \sqrt {- a \sec {\left (c + d x \right )} + a} \sec {\left (c + d x \right )} + a \sqrt {- a \sec {\left (c + d x \right )} + a}}\, dx\right ) \]
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\[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 1.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.21 \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {7 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {10 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {3 \, \sqrt {2} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A + 4 \, \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a}{{\left ({\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} + 3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a + 2 \, a^{2}\right )} a}}{2 \, d} \]
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Timed out. \[ \int \frac {\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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