Integrand size = 41, antiderivative size = 330 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\left (2 a^4-3 a^2 b^2+2 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right ) (b+a \sec (d+e x))^3}{(a-b)^{3/2} b^3 (a+b)^{3/2} e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{2 b^2 \left (a^2-b^2\right ) e \left (a^2 b+a^3 \sec (d+e x)\right ) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \]
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Time = 0.60 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4259, 4008, 4145, 4004, 3916, 2738, 211} \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )}{2 b e \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}}+\frac {x \left (a^2 \sec (d+e x)+a b\right )^3}{a^2 b^3 \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}}-\frac {\left (2 a^4-3 a^2 b^2+2 b^4\right ) (a \sec (d+e x)+b)^3 \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{b^3 e (a-b)^{3/2} (a+b)^{3/2} \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )^3}{2 b^2 e \left (a^2-b^2\right ) \left (a^3 \sec (d+e x)+a^2 b\right ) \left (a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right )^{3/2}} \]
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Rule 211
Rule 2738
Rule 3916
Rule 4004
Rule 4008
Rule 4145
Rule 4259
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a b+2 a^2 \sec (d+e x)\right )^3 \int \frac {a+b \sec (d+e x)}{\left (2 a b+2 a^2 \sec (d+e x)\right )^3} \, dx}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = -\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sec (d+e x)\right )^3 \int \frac {8 a^3 \left (a^2-b^2\right )+8 a^2 b \left (a^2-b^2\right ) \sec (d+e x)-4 a^3 \left (a^2-b^2\right ) \sec ^2(d+e x)}{\left (2 a b+2 a^2 \sec (d+e x)\right )^2} \, dx}{16 a^3 b \left (a^2-b^2\right ) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = -\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sec (d+e x)\right )^3 \int \frac {32 a^5 \left (a^2-b^2\right )^2+16 a^4 b \left (a^2-2 b^2\right ) \left (a^2-b^2\right ) \sec (d+e x)}{2 a b+2 a^2 \sec (d+e x)} \, dx}{128 a^6 b^2 \left (a^2-b^2\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = \frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (\left (-32 a^5 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )+64 a^7 \left (a^2-b^2\right )^2\right ) \left (2 a b+2 a^2 \sec (d+e x)\right )^3\right ) \int \frac {\sec (d+e x)}{2 a b+2 a^2 \sec (d+e x)} \, dx}{256 a^7 b^3 \left (a^2-b^2\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = \frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (\left (-32 a^5 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )+64 a^7 \left (a^2-b^2\right )^2\right ) \left (2 a b+2 a^2 \sec (d+e x)\right )^3\right ) \int \frac {1}{1+\frac {b \cos (d+e x)}{a}} \, dx}{512 a^9 b^3 \left (a^2-b^2\right )^2 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = \frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (\left (-32 a^5 b^2 \left (a^2-2 b^2\right ) \left (a^2-b^2\right )+64 a^7 \left (a^2-b^2\right )^2\right ) \left (2 a b+2 a^2 \sec (d+e x)\right )^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {b}{a}+\left (1-\frac {b}{a}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{256 a^9 b^3 \left (a^2-b^2\right )^2 e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ & = -\frac {\left (2 a^4-3 a^2 b^2+2 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right ) (b+a \sec (d+e x))^3}{(a-b)^{3/2} b^3 (a+b)^{3/2} e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}+\frac {x \left (a b+a^2 \sec (d+e x)\right )^3}{a^2 b^3 \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (a b+a^2 \sec (d+e x)\right ) \tan (d+e x)}{2 b e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a^2-3 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{2 a b^2 \left (a^2-b^2\right ) e \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \\ \end{align*}
Time = 2.50 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\frac {(a+b \cos (d+e x)) \sec ^2(d+e x) (a+b \sec (d+e x)) \left (2 a (d+e x) (a+b \cos (d+e x))^2+\frac {2 \left (2 a^4-3 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right ) (a+b \cos (d+e x))^2}{\left (-a^2+b^2\right )^{3/2}}+a^2 b \sin (d+e x)+\frac {a b \left (3 a^2-4 b^2\right ) (a+b \cos (d+e x)) \sin (d+e x)}{(-a+b) (a+b)}\right )}{2 b^3 e (b+a \cos (d+e x)) \left ((b+a \sec (d+e x))^2\right )^{3/2}} \]
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Result contains complex when optimal does not.
Time = 4.60 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.05
method | result | size |
risch | \(\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) a x}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, b^{3}}-\frac {i a \left (4 a^{3} b \,{\mathrm e}^{3 i \left (e x +d \right )}-5 a \,b^{3} {\mathrm e}^{3 i \left (e x +d \right )}+6 a^{4} {\mathrm e}^{2 i \left (e x +d \right )}-5 a^{2} b^{2} {\mathrm e}^{2 i \left (e x +d \right )}-4 b^{4} {\mathrm e}^{2 i \left (e x +d \right )}+8 a^{3} b \,{\mathrm e}^{i \left (e x +d \right )}-11 a \,b^{3} {\mathrm e}^{i \left (e x +d \right )}+3 a^{2} b^{2}-4 b^{4}\right )}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, b^{3} \left (a^{2}-b^{2}\right ) e}-\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) \left (2 a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) e \,b^{3}}+\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right ) \left (2 a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right ) \sqrt {\frac {\left (b \,{\mathrm e}^{2 i \left (e x +d \right )}+2 a \,{\mathrm e}^{i \left (e x +d \right )}+b \right )^{2}}{\left (1+{\mathrm e}^{2 i \left (e x +d \right )}\right )^{2}}}\, \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) e \,b^{3}}\) | \(678\) |
default | \(-\frac {2 \sin \left (e x +d \right ) \tan \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{3} b^{2} \left (e x +d \right )+2 \sqrt {\left (a +b \right ) \left (a -b \right )}\, \cos \left (e x +d \right ) a \,b^{4} \left (e x +d \right )+4 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) \cos \left (e x +d \right ) a^{4} b^{2}-6 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) \cos \left (e x +d \right ) a^{2} b^{4}+4 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) \cos \left (e x +d \right ) b^{6}+3 \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{3} b^{2} \sin \left (e x +d \right )-4 \sqrt {\left (a +b \right ) \left (a -b \right )}\, a \,b^{4} \sin \left (e x +d \right )-4 \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (e x +d \right ) a^{4} b +4 \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (e x +d \right ) a^{2} b^{3}+8 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{5} b -12 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{3} b^{3}+8 \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a \,b^{5}+2 \tan \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{4} b -3 \tan \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, a^{2} b^{3}-2 \sec \left (e x +d \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (e x +d \right ) a^{5}+4 \sec \left (e x +d \right ) \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{6}-6 \sec \left (e x +d \right ) \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{4} b^{2}+4 \sec \left (e x +d \right ) \arctan \left (\frac {\left (-\cot \left (e x +d \right )+\csc \left (e x +d \right )\right ) \left (a -b \right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) a^{2} b^{4}}{2 e \left (b \cos \left (e x +d \right )+a \right ) \sqrt {\sec \left (e x +d \right )^{2} \left (b \cos \left (e x +d \right )+a \right )^{2}}\, \sqrt {\left (a +b \right ) \left (a -b \right )}\, \left (a^{2}-b^{2}\right ) b^{3}}\) | \(695\) |
parts | \(\text {Expression too large to display}\) | \(1210\) |
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Time = 0.34 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.42 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\left [\frac {4 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} e x \cos \left (e x + d\right )^{2} + 8 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} e x \cos \left (e x + d\right ) + 4 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} e x + {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (e x + d\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (e x + d\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (e x + d\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (e x + d\right ) + b\right )} \sin \left (e x + d\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (e x + d\right )^{2} + 2 \, a b \cos \left (e x + d\right ) + a^{2}}\right ) - 2 \, {\left (2 \, a^{6} b - 5 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + {\left (3 \, a^{5} b^{2} - 7 \, a^{3} b^{4} + 4 \, a b^{6}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{4 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} e \cos \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} e \cos \left (e x + d\right ) + {\left (a^{6} b^{3} - 2 \, a^{4} b^{5} + a^{2} b^{7}\right )} e\right )}}, \frac {2 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} e x \cos \left (e x + d\right )^{2} + 4 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} e x \cos \left (e x + d\right ) + 2 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} e x - {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \left (e x + d\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (e x + d\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (e x + d\right )}\right ) - {\left (2 \, a^{6} b - 5 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + {\left (3 \, a^{5} b^{2} - 7 \, a^{3} b^{4} + 4 \, a b^{6}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} e \cos \left (e x + d\right )^{2} + 2 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} e \cos \left (e x + d\right ) + {\left (a^{6} b^{3} - 2 \, a^{4} b^{5} + a^{2} b^{7}\right )} e\right )}}\right ] \]
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\[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {a + b \sec {\left (d + e x \right )}}{\left (\left (a \sec {\left (d + e x \right )} + b\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.38 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (2 \, a^{4} - 3 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {e x + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right ) - b^{5} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right )\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 3 \, a^{3} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 4 \, a b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 2 \, a^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 3 \, a^{3} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) - 4 \, a b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )}{{\left (a^{2} b^{2} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right ) - b^{4} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right )\right )} {\left (a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + a + b\right )}^{2}} - \frac {{\left (e x + d\right )} a}{b^{3} \mathrm {sgn}\left (b \cos \left (e x + d\right )^{2} + a \cos \left (e x + d\right )\right )}}{e} \]
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Timed out. \[ \int \frac {a+b \sec (d+e x)}{\left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {a+\frac {b}{\cos \left (d+e\,x\right )}}{{\left (b^2+\frac {a^2}{{\cos \left (d+e\,x\right )}^2}+\frac {2\,a\,b}{\cos \left (d+e\,x\right )}\right )}^{3/2}} \,d x \]
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