Integrand size = 25, antiderivative size = 761 \[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}+\frac {2 \sqrt {2} \sqrt {a-b} \sqrt {a+b} e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a b d \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} \sqrt {a-b} \sqrt {a+b} e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a b d \sqrt {\sin (c+d x)}}-\frac {2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{b d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d} \]
[Out]
Time = 1.41 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.880, Rules used = {3976, 3969, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2693, 2695, 2652, 2719, 3975, 2812, 2809, 2985, 2984, 504, 1227, 551} \[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=-\frac {e^{5/2} \left (a^2-b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}+\frac {e^{5/2} \left (a^2-b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a b^2 d}+\frac {e^{5/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}-\frac {e^{5/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}+\frac {2 \sqrt {2} e^2 \sqrt {a-b} \sqrt {a+b} \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{a b d \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} e^2 \sqrt {a-b} \sqrt {a+b} \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)+1}}\right ),-1\right )}{a b d \sqrt {\sin (c+d x)}}+\frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {a e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} b^2 d}-\frac {a e^{5/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}+\frac {a e^{5/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}-\frac {2 e^2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{b d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d} \]
[In]
[Out]
Rule 210
Rule 303
Rule 335
Rule 504
Rule 551
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1227
Rule 2652
Rule 2693
Rule 2695
Rule 2719
Rule 2809
Rule 2812
Rule 2984
Rule 2985
Rule 3557
Rule 3969
Rule 3975
Rule 3976
Rubi steps \begin{align*} \text {integral}& = -\frac {e^2 \int (a-b \sec (c+d x)) \sqrt {e \tan (c+d x)} \, dx}{b^2}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \frac {\sqrt {e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx}{b^2} \\ & = -\frac {\left (a e^2\right ) \int \sqrt {e \tan (c+d x)} \, dx}{b^2}+\frac {e^2 \int \sec (c+d x) \sqrt {e \tan (c+d x)} \, dx}{b}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \sqrt {e \tan (c+d x)} \, dx}{a b^2}-\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \frac {\sqrt {e \tan (c+d x)}}{b+a \cos (c+d x)} \, dx}{a b} \\ & = \frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}-\frac {\left (2 e^2\right ) \int \cos (c+d x) \sqrt {e \tan (c+d x)} \, dx}{b}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^2 \sqrt {e \cot (c+d x)} \sqrt {e \tan (c+d x)}\right ) \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \cot (c+d x)}} \, dx}{a b} \\ & = \frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}-\frac {\left (2 a e^3\right ) \text {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{b^2 d}+\frac {\left (2 \left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^2 \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {-\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a b \sqrt {\sin (c+d x)}}-\frac {\left (2 e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{b \sqrt {\sin (c+d x)}} \\ & = \frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}+\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{b^2 d}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a b \sqrt {\sin (c+d x)}}-\frac {\left (2 e^2 \cos (c+d x) \sqrt {e \tan (c+d x)}\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{b \sqrt {\sin (2 c+2 d x)}} \\ & = -\frac {2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{b d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}-\frac {\left (a e^{5/2}\right ) \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} b^2 d}-\frac {\left (a e^{5/2}\right ) \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} b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^{5/2}\right ) \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 b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^{5/2}\right ) \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 b^2 d}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 b^2 d}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^3\right ) \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 b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^3\right ) \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 b^2 d}-\frac {\left (4 \sqrt {2} \left (a^2-b^2\right ) e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (a+b+(-a+b) x^4\right )} \, dx,x,\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right )}{a b d \sqrt {\sin (c+d x)}} \\ & = -\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{b d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}-\frac {\left (a e^{5/2}\right ) \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} b^2 d}+\frac {\left (a e^{5/2}\right ) \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} b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^{5/2}\right ) \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 b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^{5/2}\right ) \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 b^2 d}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b}-\sqrt {a-b} x^2\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right )}{a \sqrt {a-b} b d \sqrt {\sin (c+d x)}}+\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b}+\sqrt {a-b} x^2\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right )}{a \sqrt {a-b} b d \sqrt {\sin (c+d x)}} \\ & = \frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{b d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a+b}-\sqrt {a-b} x^2\right )} \, dx,x,\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right )}{a \sqrt {a-b} b d \sqrt {\sin (c+d x)}}+\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a+b}+\sqrt {a-b} x^2\right )} \, dx,x,\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right )}{a \sqrt {a-b} b d \sqrt {\sin (c+d x)}} \\ & = \frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}+\frac {2 \sqrt {2} \sqrt {a-b} \sqrt {a+b} e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a b d \sqrt {\sin (c+d x)}}-\frac {2 \sqrt {2} \sqrt {a-b} \sqrt {a+b} e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {\sqrt {a-b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {\sin (c+d x)}}{\sqrt {1+\cos (c+d x)}}\right ),-1\right ) \sqrt {e \tan (c+d x)}}{a b d \sqrt {\sin (c+d x)}}-\frac {2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{b d \sqrt {\sin (2 c+2 d x)}}+\frac {2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 52.97 (sec) , antiderivative size = 1846, normalized size of antiderivative = 2.43 \[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\frac {2 (b+a \cos (c+d x)) \cot (c+d x) (e \tan (c+d x))^{5/2}}{b d (a+b \sec (c+d x))}-\frac {(b+a \cos (c+d x)) \sec (c+d x) (e \tan (c+d x))^{5/2} \left (\frac {4 a \sec ^2(c+d x) \left (\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )+\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\tan (c+d x)}+b \tan (c+d x)\right )-\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\tan (c+d x)}+b \tan (c+d x)\right )}{4 \sqrt {2} \sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 a^2-3 b^2}\right ) \left (a+b \sqrt {1+\tan ^2(c+d x)}\right )}{(b+a \cos (c+d x)) \left (1+\tan ^2(c+d x)\right )^{3/2}}-\frac {b \sec (c+d x) \left (6 \sqrt {2} \left (a^2-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+6 \sqrt {2} b^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-(6+6 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+(6+6 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-3 \sqrt {2} a^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} b^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+3 \sqrt {2} a^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-3 \sqrt {2} b^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+(3+3 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )-(3+3 i) \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )+8 a b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)\right ) \left (a+b \sqrt {1+\tan ^2(c+d x)}\right )}{4 \left (a^3-a b^2\right ) (b+a \cos (c+d x)) \left (1+\tan ^2(c+d x)\right )}+\frac {\cos (2 (c+d x)) \sec ^2(c+d x) \left (-84 \sqrt {2} b \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+84 \sqrt {2} b \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+\frac {(42+42 i) \left (-a^2+2 b^2\right ) \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )}{\sqrt {b} \sqrt [4]{a^2-b^2}}+\frac {(42+42 i) \left (a^2-2 b^2\right ) \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )}{\sqrt {b} \sqrt [4]{a^2-b^2}}+42 \sqrt {2} b \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-42 \sqrt {2} b \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+\frac {(21+21 i) \left (a^2-2 b^2\right ) \log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )}{\sqrt {b} \sqrt [4]{a^2-b^2}}+\frac {(21+21 i) \left (-a^2+2 b^2\right ) \log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\tan (c+d x)}+i b \tan (c+d x)\right )}{\sqrt {b} \sqrt [4]{a^2-b^2}}+\frac {112 a^3 \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)}{a^2-b^2}-\frac {168 a b^2 \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {3}{2}}(c+d x)}{a^2-b^2}-\frac {24 a b^2 \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\tan ^2(c+d x),\frac {b^2 \tan ^2(c+d x)}{a^2-b^2}\right ) \tan ^{\frac {7}{2}}(c+d x)}{a^2-b^2}-\frac {168 a \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {1+\tan ^2(c+d x)}}\right ) \left (a+b \sqrt {1+\tan ^2(c+d x)}\right )}{84 a (b+a \cos (c+d x)) \left (-1+\tan ^2(c+d x)\right ) \sqrt {1+\tan ^2(c+d x)}}\right )}{b d (a+b \sec (c+d x)) \tan ^{\frac {5}{2}}(c+d x)} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2966 vs. \(2 (646 ) = 1292\).
Time = 3.69 (sec) , antiderivative size = 2967, normalized size of antiderivative = 3.90
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Timed out. \[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\int \frac {\left (e \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\int { \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \tan (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{b+a\,\cos \left (c+d\,x\right )} \,d x \]
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