Integrand size = 33, antiderivative size = 480 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (3 a^3 A b-9 a A b^3-15 a^4 B+29 a^2 b^2 B-8 b^4 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 A b-7 A b^3-5 a^3 B+11 a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^4 A b-6 a^2 A b^3+15 A b^5-15 a^5 B+38 a^3 b^2 B-35 a b^4 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^3 (a+b)^3 d}-\frac {\left (3 a^3 A b-9 a A b^3-15 a^4 B+29 a^2 b^2 B-8 b^4 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (a^2 A b-7 A b^3-5 a^3 B+11 a b^2 B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
1/2*a*(A*b-B*a)*sec(d*x+c)^(5/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c)) ^2+1/4*a*(A*a^2*b-7*A*b^3-5*B*a^3+11*B*a*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/ b^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))-1/4*(3*A*a^3*b-9*A*a*b^3-15*B*a^4+29*B* a^2*b^2-8*B*b^4)*sin(d*x+c)*sec(d*x+c)^(1/2)/b^3/(a^2-b^2)^2/d+1/4*(3*A*a^ 3*b-9*A*a*b^3-15*B*a^4+29*B*a^2*b^2-8*B*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/ cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)* sec(d*x+c)^(1/2)/b^3/(a^2-b^2)^2/d+1/4*(A*a^2*b-7*A*b^3-5*B*a^3+11*B*a*b^2 )*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/ 2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/b^2/(a^2-b^2)^2/d+1/4*(3*A *a^4*b-6*A*a^2*b^3+15*A*b^5-15*B*a^5+38*B*a^3*b^2-35*B*a*b^4)*(cos(1/2*d*x +1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b ),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/(a-b)^2/b^3/(a+b)^3/d
Time = 8.23 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.75 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=-\frac {\frac {2 \left (-9 a^4 A b+19 a^2 A b^3-16 A b^5+45 a^5 B-95 a^3 b^2 B+56 a b^4 B\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-8 a^3 A b^2+32 a A b^4+40 a^4 b B-80 a^2 b^3 B+16 b^5 B\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-3 a^4 A b+9 a^2 A b^3+15 a^5 B-29 a^3 b^2 B+8 a b^4 B\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{16 (a-b)^2 b^3 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {\left (-3 a^3 A b+9 a A b^3+15 a^4 B-29 a^2 b^2 B+8 b^4 B\right ) \sin (c+d x)}{4 b^3 \left (-a^2+b^2\right )^2}+\frac {-a A b \sin (c+d x)+a^2 B \sin (c+d x)}{2 b \left (-a^2+b^2\right ) (b+a \cos (c+d x))^2}+\frac {a^3 A b \sin (c+d x)-7 a A b^3 \sin (c+d x)-5 a^4 B \sin (c+d x)+11 a^2 b^2 B \sin (c+d x)}{4 b^2 \left (-a^2+b^2\right )^2 (b+a \cos (c+d x))}\right )}{d} \]
-1/16*((2*(-9*a^4*A*b + 19*a^2*A*b^3 - 16*A*b^5 + 45*a^5*B - 95*a^3*b^2*B + 56*a*b^4*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*S qrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-8*a^3*A*b^2 + 32*a*A*b^4 + 40*a^4*b*B - 80*a^2*b^3*B + 16 *b^5*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]* (a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[ c + d*x])*(1 - Cos[c + d*x]^2)) + ((-3*a^4*A*b + 9*a^2*A*b^3 + 15*a^5*B - 29*a^3*b^2*B + 8*a*b^4*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqr t[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[ Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a ^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*S qrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x ]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b *(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-3*a^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*B)*Sin[c + d*x])/(4*b^3*(-a ^2 + b^2)^2) + (-(a*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x])/(2*b*(-a^2 + b ^2)*(b + a*Cos[c + d*x])^2) + (a^3*A*b*Sin[c + d*x] - 7*a*A*b^3*Sin[c +...
Time = 3.38 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4517, 27, 3042, 4586, 27, 3042, 4590, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4517 |
| \(\displaystyle \frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-5 B a^2+A b a+4 b^2 B\right ) \sec ^2(c+d x)\right )-4 b (A b-a B) \sec (c+d x)+3 a (A b-a B)\right )}{2 (a+b \sec (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-5 B a^2+A b a+4 b^2 B\right ) \sec ^2(c+d x)\right )-4 b (A b-a B) \sec (c+d x)+3 a (A b-a B)\right )}{(a+b \sec (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (5 B a^2-A b a-4 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-4 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (A b-a B)\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle \frac {\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\sqrt {\sec (c+d x)} \left (-\left (\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) \sec ^2(c+d x)\right )+4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sec (c+d x)+a \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right )\right )}{2 (a+b \sec (c+d x))}dx}{b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (-\left (\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) \sec ^2(c+d x)\right )+4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \sec (c+d x)+a \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right )\right )}{a+b \sec (c+d x)}dx}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (15 B a^4-3 A b a^3-29 b^2 B a^2+9 A b^3 a+8 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (B a^3+A b a^2-4 b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) \sec ^2(c+d x)+4 b \left (-5 B a^4+A b a^3+10 b^2 B a^2-4 A b^3 a-2 b^4 B\right ) \sec (c+d x)+a \left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) \sec ^2(c+d x)+4 b \left (-5 B a^4+A b a^3+10 b^2 B a^2-4 A b^3 a-2 b^4 B\right ) \sec (c+d x)+a \left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (-15 B a^5+3 A b a^4+33 b^2 B a^3-5 A b^3 a^2-24 b^4 B a+8 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (-5 B a^4+A b a^3+10 b^2 B a^2-4 A b^3 a-2 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) a^2+b \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \sec (c+d x) a^2}{\sqrt {\sec (c+d x)}}dx}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (-15 B a^4+3 A b a^3+29 b^2 B a^2-9 A b^3 a-8 b^4 B\right ) a^2+b \left (-5 B a^3+A b a^2+11 b^2 B a-7 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \int \sqrt {\sec (c+d x)}dx+a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}+\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle \frac {\frac {\frac {\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx+\frac {\frac {2 a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {2 a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {a \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {\frac {\frac {2 a^2 b \left (-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 a^2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}+\frac {2 \left (-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}-\frac {2 \left (-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
(a*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b* Sec[c + d*x])^2) + ((a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) + ((((2*a^ 2*(3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*b*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/ 2, 2]*Sqrt[Sec[c + d*x]])/d)/a^2 + (2*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a )/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d))/b - (2*(3*a^3* A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Sec[c + d*x]]*Si n[c + d*x])/(b*d))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
3.5.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*d^2*( A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[d/(b*(m + 1)*(a^2 - b^2)) Int[( a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*d*(A*b - a*B)*( n - 2) + b*d*(A*b - a*B)*(m + 1)*Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2 *(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f , A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[ n, 1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) ), x] + Simp[d/(b*(a^2 - b^2)*(m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*( d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C }, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2) Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2 Int[(a*A - (A*b - a *B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1996\) vs. \(2(528)=1056\).
Time = 199.88 (sec) , antiderivative size = 1997, normalized size of antiderivative = 4.16
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B/b^3/sin(1/ 2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 *d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/2*d *x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/ 2*c)^2-1)^(1/2))-2*B*a/b^2*(a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2 *d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)-1 /2/(a+b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/ (-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x +1/2*c),2^(1/2))+1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/ 2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/ 2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+s in(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/b/(a^ 2-b^2)/(a^2-a*b)*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi( cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^2-a*b)*a*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2 *c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b), 2^(1/2)))+2*B*a^2/b^3/(a^2-a*b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d *x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/...
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]