Integrand size = 25, antiderivative size = 339 \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {e^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{11/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d} \]
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Time = 0.64 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3973, 3971, 3554, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2691, 2694, 2653, 2720, 2687, 32} \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {e^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{11/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d} \]
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Rule 32
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2653
Rule 2687
Rule 2691
Rule 2694
Rule 2720
Rule 3554
Rule 3557
Rule 3971
Rule 3973
Rubi steps \begin{align*} \text {integral}& = \frac {e^4 \int (-a+a \sec (c+d x))^2 (e \tan (c+d x))^{3/2} \, dx}{a^4} \\ & = \frac {e^4 \int \left (a^2 (e \tan (c+d x))^{3/2}-2 a^2 \sec (c+d x) (e \tan (c+d x))^{3/2}+a^2 \sec ^2(c+d x) (e \tan (c+d x))^{3/2}\right ) \, dx}{a^4} \\ & = \frac {e^4 \int (e \tan (c+d x))^{3/2} \, dx}{a^2}+\frac {e^4 \int \sec ^2(c+d x) (e \tan (c+d x))^{3/2} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \sec (c+d x) (e \tan (c+d x))^{3/2} \, dx}{a^2} \\ & = \frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {e^4 \text {Subst}\left (\int (e x)^{3/2} \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {\left (2 e^6\right ) \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{3 a^2}-\frac {e^6 \int \frac {1}{\sqrt {e \tan (c+d x)}} \, dx}{a^2} \\ & = \frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {e^7 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \tan (c+d x)\right )}{a^2 d}+\frac {\left (2 e^6 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 a^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}} \\ & = \frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {\left (2 e^7\right ) \text {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}+\frac {\left (2 e^6 \sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{3 a^2 \sqrt {e \tan (c+d x)}} \\ & = \frac {2 e^6 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {e^6 \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}-\frac {e^6 \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d} \\ & = \frac {2 e^6 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}+\frac {e^{11/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {e^{11/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^6 \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a^2 d}-\frac {e^6 \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a^2 d} \\ & = \frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d}-\frac {e^{11/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d} \\ & = \frac {e^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}-\frac {e^{11/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d}+\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}-\frac {e^{11/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d}+\frac {2 e^6 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a^2 d \sqrt {e \tan (c+d x)}}+\frac {2 e^5 \sqrt {e \tan (c+d x)}}{a^2 d}-\frac {4 e^5 \sec (c+d x) \sqrt {e \tan (c+d x)}}{3 a^2 d}+\frac {2 e^3 (e \tan (c+d x))^{5/2}}{5 a^2 d} \\ \end{align*}
\[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx \]
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Result contains complex when optimal does not.
Time = 4.51 (sec) , antiderivative size = 1007, normalized size of antiderivative = 2.97
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Timed out. \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \tan (c+d x))^{11/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{11/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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