Integrand size = 35, antiderivative size = 513 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {\left (472 a^2 A b+128 A b^3+133 a^3 B+356 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{192 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 A b+160 a A b^3-5 a^4 B+120 a^2 b^2 B+48 b^4 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{64 b d \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (264 a^2 A b+128 A b^3+15 a^3 B+284 a b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (104 a A b+59 a^2 B+36 b^2 B\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {b (8 A b+11 a B) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \]
1/4*b*B*sec(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/192*(472*A* a^2*b+128*A*b^3+133*B*a^3+356*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)*(a/(a+b))^(1/2))*((b+a*c os(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/64*(40 *A*a^3*b+160*A*a*b^3-5*B*a^4+120*B*a^2*b^2+48*B*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),2,2^(1/2)*(a/(a+b ))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/b/d/(a+b*sec(d*x +c))^(1/2)+1/96*(104*A*a*b+59*B*a^2+36*B*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)* (a+b*sec(d*x+c))^(1/2)/d+1/24*b*(8*A*b+11*B*a)*sec(d*x+c)^(5/2)*sin(d*x+c) *(a+b*sec(d*x+c))^(1/2)/d-1/192*(264*A*a^2*b+128*A*b^3+15*B*a^3+284*B*a*b^ 2)*(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)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/b/d/((b+a*cos(d*x+c) )/(a+b))^(1/2)/sec(d*x+c)^(1/2)+1/192*(264*A*a^2*b+128*A*b^3+15*B*a^3+284* B*a*b^2)*sin(d*x+c)*sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/b/d
Result contains complex when optimal does not.
Time = 7.90 (sec) , antiderivative size = 768, normalized size of antiderivative = 1.50 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=-\frac {(a+b \sec (c+d x))^{5/2} \left (\frac {2 \left (-416 a^2 A b^2-236 a^3 b B-144 a b^3 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (24 a^3 A b-832 a A b^3+45 a^4 B-436 a^2 b^2 B-288 b^4 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (264 a^3 A b+128 a A b^3+15 a^4 B+284 a^2 b^2 B\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{768 b d (b+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (\frac {1}{24} \sec ^3(c+d x) \left (8 A b^2 \sin (c+d x)+17 a b B \sin (c+d x)\right )+\frac {1}{96} \sec ^2(c+d x) \left (104 a A b \sin (c+d x)+59 a^2 B \sin (c+d x)+36 b^2 B \sin (c+d x)\right )+\frac {\sec (c+d x) \left (264 a^2 A b \sin (c+d x)+128 A b^3 \sin (c+d x)+15 a^3 B \sin (c+d x)+284 a b^2 B \sin (c+d x)\right )}{192 b}+\frac {1}{4} b^2 B \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \]
-1/768*((a + b*Sec[c + d*x])^(5/2)*((2*(-416*a^2*A*b^2 - 236*a^3*b*B - 144 *a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/ (a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(24*a^3*A*b - 832*a*A*b^3 + 45*a^4 *B - 436*a^2*b^2*B - 288*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Ellipti cPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(264 *a^3*A*b + 128*a*A*b^3 + 15*a^4*B + 284*a^2*b^2*B)*Sqrt[(a - a*Cos[c + d*x ])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*C os[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/( Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x] ^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x]) ^2))))/(b*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((Sec[c + d*x]^3*(8*A*b^2*Sin[c + d*x] + 17*a*b*B*Sin[c + d *x]))/24 + (Sec[c + d*x]^2*(104*a*A*b*Sin[c + d*x] + 59*a^2*B*Sin[c + d*x] + 36*b^2*B*Sin[c + d*x]))/96 + (Sec[c + d*x]*(264*a^2*A*b*Sin[c + d*x] + 128*A*b^3*Sin[c + d*x] + 15*a^3*B*Sin[c + d*x] + 284*a*b^2*B*Sin[c + d*x]) )/(192*b) + (b^2*B*Sec[c + d*x]^3*Tan[c + d*x])/4))/(d*(b + a*Cos[c + d*x] )^2*Sec[c + d*x]^(5/2))
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.
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4514 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (8 A b+11 a B) \sec ^2(c+d x)+2 \left (4 B a^2+8 A b a+3 b^2 B\right ) \sec (c+d x)+a (8 a A+3 b B)\right )dx+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (8 A b+11 a B) \sec ^2(c+d x)+2 \left (4 B a^2+8 A b a+3 b^2 B\right ) \sec (c+d x)+a (8 a A+3 b B)\right )dx+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (8 A b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (4 B a^2+8 A b a+3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (8 a A+3 b B)\right )dx+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (b \left (59 B a^2+104 A b a+36 b^2 B\right ) \sec ^2(c+d x)+2 \left (24 B a^3+72 A b a^2+49 b^2 B a+16 A b^3\right ) \sec (c+d x)+3 a \left (16 A a^2+17 b B a+8 A b^2\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (b \left (59 B a^2+104 A b a+36 b^2 B\right ) \sec ^2(c+d x)+2 \left (24 B a^3+72 A b a^2+49 b^2 B a+16 A b^3\right ) \sec (c+d x)+3 a \left (16 A a^2+17 b B a+8 A b^2\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (24 B a^3+72 A b a^2+49 b^2 B a+16 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (16 A a^2+17 b B a+8 A b^2\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right ) \sec ^2(c+d x)+2 b \left (96 A a^3+161 b B a^2+152 A b^2 a+36 b^3 B\right ) \sec (c+d x)+a b \left (59 B a^2+104 A b a+36 b^2 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right ) \sec ^2(c+d x)+2 b \left (96 A a^3+161 b B a^2+152 A b^2 a+36 b^3 B\right ) \sec (c+d x)+a b \left (59 B a^2+104 A b a+36 b^2 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (96 A a^3+161 b B a^2+152 A b^2 a+36 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a b \left (59 B a^2+104 A b a+36 b^2 B\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int -\frac {-2 a \left (59 B a^2+104 A b a+36 b^2 B\right ) \sec (c+d x) b^2-3 \left (-5 B a^4+40 A b a^3+120 b^2 B a^2+160 A b^3 a+48 b^4 B\right ) \sec ^2(c+d x) b+a \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right ) b}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}+\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {-2 a \left (59 B a^2+104 A b a+36 b^2 B\right ) \sec (c+d x) b^2-3 \left (-5 B a^4+40 A b a^3+120 b^2 B a^2+160 A b^3 a+48 b^4 B\right ) \sec ^2(c+d x) b+a \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right ) b}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {-2 a \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b^2-3 \left (-5 B a^4+40 A b a^3+120 b^2 B a^2+160 A b^3 a+48 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b+a \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right ) b}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-3 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-3 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\right )-2 a b^2 \left (59 B a^2+104 A b a+36 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx\right )+b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\left (59 a^2 B+104 a A b+36 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
3.5.48.3.1 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[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], 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[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[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_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[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[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(m + n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n* Simp[a^2*A*(m + n) + a*b*B*n + (a*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1) )*Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2, x], x] , x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && !(IGtQ[n, 1] && !IntegerQ[m])
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 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)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a *B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[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[(-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_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[C/d^2 Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*C sc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[ a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 32.18 (sec) , antiderivative size = 7185, normalized size of antiderivative = 14.01
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(7185\) |
| default | \(\text {Expression too large to display}\) | \(7213\) |
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]