Integrand size = 25, antiderivative size = 1101 \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\frac {5 b^3 \sqrt [4]{a^2-b^2} e^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{11/2} d}-\frac {2 b \left (a^2-b^2\right )^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{11/2} d}+\frac {5 b^3 \sqrt [4]{a^2-b^2} e^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{11/2} d}-\frac {2 b \left (a^2-b^2\right )^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{11/2} d}+\frac {10 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {4 b^2 \left (4 a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \left (a^2-b^2\right ) e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^6 \left (a^2-b^2-a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b^2 \left (a^2-b^2\right )^2 e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^6 \left (a^2-b^2-a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \left (a^2-b^2\right ) e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^6 \left (a^2-b^2+a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b^2 \left (a^2-b^2\right )^2 e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^6 \left (a^2-b^2+a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))} \]
[Out]
Time = 3.52 (sec) , antiderivative size = 1101, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3957, 2991, 2715, 2721, 2720, 2772, 2944, 2946, 2781, 2886, 2884, 335, 218, 214, 211, 2774} \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=-\frac {5 b^2 \left (a^2-3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {4 b^2 \left (4 a^2-3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{3 a^6 d \sqrt {e \sin (c+d x)}}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{21 a^2 d \sqrt {e \sin (c+d x)}}+\frac {2 b^2 \left (a^2-b^2\right )^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{a^6 \left (a^2-\sqrt {a^2-b^2} a-b^2\right ) d \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \left (a^2-b^2\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{2 a^6 \left (a^2-\sqrt {a^2-b^2} a-b^2\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b^2 \left (a^2-b^2\right )^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{a^6 \left (a^2+\sqrt {a^2-b^2} a-b^2\right ) d \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \left (a^2-b^2\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^4}{2 a^6 \left (a^2+\sqrt {a^2-b^2} a-b^2\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b \left (a^2-b^2\right )^{5/4} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{7/2}}{a^{11/2} d}+\frac {5 b^3 \sqrt [4]{a^2-b^2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{7/2}}{2 a^{11/2} d}-\frac {2 b \left (a^2-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{7/2}}{a^{11/2} d}+\frac {5 b^3 \sqrt [4]{a^2-b^2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{7/2}}{2 a^{11/2} d}-\frac {10 \cos (c+d x) \sqrt {e \sin (c+d x)} e^3}{21 a^2 d}-\frac {5 b^2 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)} e^3}{3 a^5 d}+\frac {4 b \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)} e^3}{3 a^5 d}-\frac {2 \cos (c+d x) (e \sin (c+d x))^{5/2} e}{7 a^2 d}+\frac {4 b (e \sin (c+d x))^{5/2} e}{5 a^3 d}+\frac {b^2 (e \sin (c+d x))^{5/2} e}{a^3 d (b+a \cos (c+d x))} \]
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 335
Rule 2715
Rule 2720
Rule 2721
Rule 2772
Rule 2774
Rule 2781
Rule 2884
Rule 2886
Rule 2944
Rule 2946
Rule 2991
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{7/2}}{(-b-a \cos (c+d x))^2} \, dx \\ & = \int \left (\frac {(e \sin (c+d x))^{7/2}}{a^2}+\frac {b^2 (e \sin (c+d x))^{7/2}}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b (e \sin (c+d x))^{7/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx \\ & = \frac {\int (e \sin (c+d x))^{7/2} \, dx}{a^2}-\frac {(2 b) \int \frac {(e \sin (c+d x))^{7/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac {b^2 \int \frac {(e \sin (c+d x))^{7/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2} \\ & = \frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (5 e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx}{7 a^2}+\frac {\left (2 b e^2\right ) \int \frac {(-a-b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac {\left (5 b^2 e^2\right ) \int \frac {\cos (c+d x) (e \sin (c+d x))^{3/2}}{b+a \cos (c+d x)} \, dx}{2 a^3} \\ & = -\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (5 e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 a^2}+\frac {\left (4 b e^4\right ) \int \frac {-\frac {1}{2} a \left (3 a^2-2 b^2\right )-\frac {1}{2} b \left (4 a^2-3 b^2\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 a^5}-\frac {\left (5 b^2 e^4\right ) \int \frac {-a b+\frac {1}{2} \left (a^2-3 b^2\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 a^5} \\ & = -\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (5 b^2 \left (a^2-3 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{6 a^6}-\frac {\left (2 b^2 \left (4 a^2-3 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a^6}+\frac {\left (5 b^3 \left (a^2-b^2\right ) e^4\right ) \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{2 a^6}-\frac {\left (2 b \left (a^2-b^2\right )^2 e^4\right ) \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{a^6}+\frac {\left (5 e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a^2 \sqrt {e \sin (c+d x)}} \\ & = \frac {10 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (5 b^4 \sqrt {a^2-b^2} e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^6}-\frac {\left (5 b^4 \sqrt {a^2-b^2} e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^6}+\frac {\left (b^2 \left (a^2-b^2\right )^{3/2} e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^6}+\frac {\left (b^2 \left (a^2-b^2\right )^{3/2} e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^6}-\frac {\left (5 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{2 a^5 d}+\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{a^5 d}-\frac {\left (5 b^2 \left (a^2-3 b^2\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{6 a^6 \sqrt {e \sin (c+d x)}}-\frac {\left (2 b^2 \left (4 a^2-3 b^2\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^6 \sqrt {e \sin (c+d x)}} \\ & = \frac {10 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {4 b^2 \left (4 a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (5 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}+\frac {\left (4 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}-\frac {\left (5 b^4 \sqrt {a^2-b^2} e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^6 \sqrt {e \sin (c+d x)}}-\frac {\left (5 b^4 \sqrt {a^2-b^2} e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^6 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \left (a^2-b^2\right )^{3/2} e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^6 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \left (a^2-b^2\right )^{3/2} e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^6 \sqrt {e \sin (c+d x)}} \\ & = \frac {10 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {4 b^2 \left (4 a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}+\frac {5 b^4 \sqrt {a^2-b^2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^6 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^6 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \sqrt {a^2-b^2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^6 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^6 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (5 b^3 \sqrt {a^2-b^2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{2 a^5 d}+\frac {\left (5 b^3 \sqrt {a^2-b^2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{2 a^5 d}-\frac {\left (2 b \left (a^2-b^2\right )^{3/2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}-\frac {\left (2 b \left (a^2-b^2\right )^{3/2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d} \\ & = \frac {5 b^3 \sqrt [4]{a^2-b^2} e^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{11/2} d}-\frac {2 b \left (a^2-b^2\right )^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{11/2} d}+\frac {5 b^3 \sqrt [4]{a^2-b^2} e^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{11/2} d}-\frac {2 b \left (a^2-b^2\right )^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{11/2} d}+\frac {10 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}-\frac {4 b^2 \left (4 a^2-3 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^6 d \sqrt {e \sin (c+d x)}}+\frac {5 b^4 \sqrt {a^2-b^2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^6 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^6 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \sqrt {a^2-b^2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^6 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^6 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}-\frac {5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 a^5 d}+\frac {4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 16.73 (sec) , antiderivative size = 2095, normalized size of antiderivative = 1.90 \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\text {Result too large to show} \]
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Time = 45.23 (sec) , antiderivative size = 1494, normalized size of antiderivative = 1.36
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\left (b+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]
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