Integrand size = 35, antiderivative size = 650 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+1617 b^5 (13 A+11 C)+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{45045 b^4 d}+\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^3 d}-\frac {2 \left (90 a^4 C-539 b^4 (13 A+11 C)-15 a^2 b^2 (715 A+543 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{45045 b^2 d}+\frac {2 a \left (2717 A b^2+15 a^2 C+2209 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9009 b d}+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac {10 a C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{13 d} \] Output:
2/45045*(a-b)*(a+b)^(1/2)*(240*a^6*C-1617*b^6*(13*A+11*C)+10*a^4*b^2*(143* A+76*C)-3*a^2*b^4*(13299*A+10223*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c)) ^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b *(1+sec(d*x+c))/(a-b))^(1/2)/b^5/d+2/45045*(a-b)*(a+b)^(1/2)*(240*a^5*C+18 0*a^4*b*C+1617*b^5*(13*A+11*C)+10*a^3*b^2*(143*A+94*C)+15*a^2*b^3*(1573*A+ 1175*C)-6*a*b^4*(2717*A+2174*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/ 2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+ sec(d*x+c))/(a-b))^(1/2)/b^4/d+2/45045*a*(120*a^4*C+5*a^2*b^2*(143*A+79*C) +b^4*(23309*A+18973*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d-2/45045*(9 0*a^4*C-539*b^4*(13*A+11*C)-15*a^2*b^2*(715*A+543*C))*sec(d*x+c)*(a+b*sec( d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/9009*a*(2717*A*b^2+15*C*a^2+2209*C*b^2)*s ec(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b/d+2/1287*(15*C*a^2+11*b^2* (13*A+11*C))*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d+10/143*a*C*s ec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/13*C*sec(d*x+c)^3*(a+b*s ec(d*x+c))^(5/2)*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(4551\) vs. \(2(650)=1300\).
Time = 24.14 (sec) , antiderivative size = 4551, normalized size of antiderivative = 7.00 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2) ,x]
Output:
(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*(-14 30*a^4*A*b^2 + 39897*a^2*A*b^4 + 21021*A*b^6 - 240*a^6*C - 760*a^4*b^2*C + 30669*a^2*b^4*C + 17787*b^6*C)*Sin[c + d*x])/(45045*b^4) + (4*Sec[c + d*x ]^4*(143*A*b^2*Sin[c + d*x] + 159*a^2*C*Sin[c + d*x] + 121*b^2*C*Sin[c + d *x]))/1287 + (4*Sec[c + d*x]^3*(2717*a*A*b^2*Sin[c + d*x] + 15*a^3*C*Sin[c + d*x] + 2209*a*b^2*C*Sin[c + d*x]))/(9009*b) + (4*Sec[c + d*x]^2*(10725* a^2*A*b^2*Sin[c + d*x] + 7007*A*b^4*Sin[c + d*x] - 90*a^4*C*Sin[c + d*x] + 8145*a^2*b^2*C*Sin[c + d*x] + 5929*b^4*C*Sin[c + d*x]))/(45045*b^2) + (4* Sec[c + d*x]*(715*a^3*A*b^2*Sin[c + d*x] + 23309*a*A*b^4*Sin[c + d*x] + 12 0*a^5*C*Sin[c + d*x] + 395*a^3*b^2*C*Sin[c + d*x] + 18973*a*b^4*C*Sin[c + d*x]))/(45045*b^3) + (108*a*b*C*Sec[c + d*x]^4*Tan[c + d*x])/143 + (4*b^2* C*Sec[c + d*x]^5*Tan[c + d*x])/13))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A *Cos[2*c + 2*d*x])) + (4*((4*a^4*A)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Se c[c + d*x]]) - (62*a^2*A*b)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x] ]) - (14*A*b^3)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^6 *C)/(3003*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (304*a^4*C)/( 9009*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (20446*a^2*b*C)/(150 15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (154*b^3*C)/(195*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (248*a^3*A*Sqrt[Sec[c + d*x]])/(31 5*Sqrt[b + a*Cos[c + d*x]]) + (4*a^5*A*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt...
Time = 3.92 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.05, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.657, Rules used = {3042, 4585, 27, 3042, 4584, 27, 3042, 4584, 27, 3042, 4590, 27, 3042, 4580, 27, 3042, 4570, 27, 3042, 4493, 3042, 4319, 4492}
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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4585 |
| \(\displaystyle \frac {2}{13} \int \frac {1}{2} \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (5 a C \sec ^2(c+d x)+b (13 A+11 C) \sec (c+d x)+a (13 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (5 a C \sec ^2(c+d x)+b (13 A+11 C) \sec (c+d x)+a (13 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (13 A+11 C) \csc \left (c+d x+\frac {\pi }{2}\right )+a (13 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left ((143 A+96 C) a^2+2 b (143 A+116 C) \sec (c+d x) a+\left (15 C a^2+11 b^2 (13 A+11 C)\right ) \sec ^2(c+d x)\right )dx+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left ((143 A+96 C) a^2+2 b (143 A+116 C) \sec (c+d x) a+\left (15 C a^2+11 b^2 (13 A+11 C)\right ) \sec ^2(c+d x)\right )dx+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left ((143 A+96 C) a^2+2 b (143 A+116 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+\left (15 C a^2+11 b^2 (13 A+11 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {\sec ^3(c+d x) \left (a \left (15 C a^2+2717 A b^2+2209 b^2 C\right ) \sec ^2(c+d x)+b \left ((3861 A+3057 C) a^2+77 b^2 (13 A+11 C)\right ) \sec (c+d x)+3 a \left (3 (143 A+106 C) a^2+22 b^2 (13 A+11 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sec ^3(c+d x) \left (a \left (15 C a^2+2717 A b^2+2209 b^2 C\right ) \sec ^2(c+d x)+b \left ((3861 A+3057 C) a^2+77 b^2 (13 A+11 C)\right ) \sec (c+d x)+3 a \left (3 (143 A+106 C) a^2+22 b^2 (13 A+11 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \left (15 C a^2+2717 A b^2+2209 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left ((3861 A+3057 C) a^2+77 b^2 (13 A+11 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (3 (143 A+106 C) a^2+22 b^2 (13 A+11 C)\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \int \frac {\sec ^2(c+d x) \left (4 \left (15 C a^2+2717 A b^2+2209 b^2 C\right ) a^2+b \left ((9009 A+6753 C) a^2+b^2 (19591 A+16127 C)\right ) \sec (c+d x) a-\left (90 C a^4-15 b^2 (715 A+543 C) a^2-539 b^4 (13 A+11 C)\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {\sec ^2(c+d x) \left (4 \left (15 C a^2+2717 A b^2+2209 b^2 C\right ) a^2+b \left ((9009 A+6753 C) a^2+b^2 (19591 A+16127 C)\right ) \sec (c+d x) a-\left (90 C a^4-15 b^2 (715 A+543 C) a^2-539 b^4 (13 A+11 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (4 \left (15 C a^2+2717 A b^2+2209 b^2 C\right ) a^2+b \left ((9009 A+6753 C) a^2+b^2 (19591 A+16127 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+\left (-90 C a^4+15 b^2 (715 A+543 C) a^2+539 b^4 (13 A+11 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {2 \int -\frac {\sec (c+d x) \left (-3 a \left (120 C a^4+5 b^2 (143 A+79 C) a^2+b^4 (23309 A+18973 C)\right ) \sec ^2(c+d x)-b \left (30 C a^4+5 b^2 (17303 A+13723 C) a^2+1617 b^4 (13 A+11 C)\right ) \sec (c+d x)+2 a \left (90 C a^4-15 b^2 (715 A+543 C) a^2-539 b^4 (13 A+11 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\int \frac {\sec (c+d x) \left (-3 a \left (120 C a^4+5 b^2 (143 A+79 C) a^2+b^4 (23309 A+18973 C)\right ) \sec ^2(c+d x)-b \left (30 C a^4+5 b^2 (17303 A+13723 C) a^2+1617 b^4 (13 A+11 C)\right ) \sec (c+d x)+2 a \left (90 C a^4-15 b^2 (715 A+543 C) a^2-539 b^4 (13 A+11 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-3 a \left (120 C a^4+5 b^2 (143 A+79 C) a^2+b^4 (23309 A+18973 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-b \left (30 C a^4+5 b^2 (17303 A+13723 C) a^2+1617 b^4 (13 A+11 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left (90 C a^4-15 b^2 (715 A+543 C) a^2-539 b^4 (13 A+11 C)\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\frac {2 \int \frac {3 \sec (c+d x) \left (a b \left (60 C a^4-5 b^2 (4433 A+3337 C) a^2-3 b^4 (12441 A+10277 C)\right )+\left (240 C a^6+10 b^2 (143 A+76 C) a^4-3 b^4 (13299 A+10223 C) a^2-1617 b^6 (13 A+11 C)\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\frac {\int \frac {\sec (c+d x) \left (a b \left (60 C a^4-5 b^2 (4433 A+3337 C) a^2-3 b^4 (12441 A+10277 C)\right )+\left (240 C a^6+10 b^2 (143 A+76 C) a^4-3 b^4 (13299 A+10223 C) a^2-1617 b^6 (13 A+11 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a b \left (60 C a^4-5 b^2 (4433 A+3337 C) a^2-3 b^4 (12441 A+10277 C)\right )+\left (240 C a^6+10 b^2 (143 A+76 C) a^4-3 b^4 (13299 A+10223 C) a^2-1617 b^6 (13 A+11 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx-(a-b) \left (240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-(a-b) \left (240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {-\frac {\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (15 a^2 C+11 b^2 (13 A+11 C)\right ) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}+\frac {1}{9} \left (\frac {2 a \left (15 a^2 C+2717 A b^2+2209 b^2 C\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}+\frac {-\frac {2 \left (90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}-\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}-\frac {2 a \left (120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}}{7 b}\right )\right )+\frac {10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}\) |
Input:
Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]
Output:
(2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d) + ((10 *a*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d) + ((2* (15*a^2*C + 11*b^2*(13*A + 11*C))*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]* Tan[c + d*x])/(9*d) + ((2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*Sec[c + d *x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d) + ((-2*(90*a^4*C - 53 9*b^4*(13*A + 11*C) - 15*a^2*b^2*(715*A + 543*C))*Sec[c + d*x]*Sqrt[a + b* Sec[c + d*x]]*Tan[c + d*x])/(5*b*d) - (((-2*(a - b)*Sqrt[a + b]*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*(a - b)*Sqrt[a + b]*(240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) + 10*a^3*b^2*(143*A + 94*C) + 15*a^2 *b^3*(1573*A + 1175*C) - 6*a*b^4*(2717*A + 2174*C))*Cot[c + d*x]*EllipticF [ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)) /b - (2*a*(120*a^4*C + 5*a^2*b^2*(143*A + 79*C) + b^4*(23309*A + 18973*C)) *Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(b*d))/(5*b))/(7*b))/9)/11)/13
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(A - B) Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[B Int[Csc[e + f*x]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B} , x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ (e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x _Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* (m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & & NeQ[a^2 - b^2, 0] && !LtQ[m, -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)*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_)]^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*Csc[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 + b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x ] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(3399\) vs. \(2(604)=1208\).
Time = 622.24 (sec) , antiderivative size = 3400, normalized size of antiderivative = 5.23
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3400\) |
| default | \(\text {Expression too large to display}\) | \(3401\) |
Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x,method=_RETUR NVERBOSE)
Output:
-2/315*A/d/b^2*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a+a*cos(d*x+c)+b*cos(d *x+c)+b)*(10*(cos(d*x+c)^2+2*cos(d*x+c)+1)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/ 2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^5*EllipticE(-csc(d*x+ c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+10*(cos(d*x+c)^2+2*cos(d*x+c)+1)*(cos(d *x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2 )*a^4*b*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+279*(-cos(d* x+c)^2-2*cos(d*x+c)-1)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos (d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c),(( a-b)/(a+b))^(1/2))+279*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*(cos(d*x+c)/(cos(d*x +c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^3*Elli pticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+147*(-cos(d*x+c)^2-2*cos (d*x+c)-1)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(co s(d*x+c)+1))^(1/2)*a*b^4*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1 /2))+147*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)* (1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^5*EllipticE(-csc(d*x+c)+ cot(d*x+c),((a-b)/(a+b))^(1/2))+10*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*(cos(d*x +c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)* a^4*b*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+155*(cos(d*x+c )^2+2*cos(d*x+c)+1)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d* x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticF(-csc(d*x+c)+cot(d*x+c),((...
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algori thm="fricas")
Output:
integral((C*b^2*sec(d*x + c)^7 + 2*C*a*b*sec(d*x + c)^6 + 2*A*a*b*sec(d*x + c)^4 + A*a^2*sec(d*x + c)^3 + (C*a^2 + A*b^2)*sec(d*x + c)^5)*sqrt(b*sec (d*x + c) + a), x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(5/2)*(A+C*sec(d*x+c)**2),x)
Output:
Timed out
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algori thm="maxima")
Output:
Timed out
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algori thm="giac")
Output:
integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^3 , x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(5/2))/cos(c + d*x)^3,x)
Output:
int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(5/2))/cos(c + d*x)^3, x)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{7}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{6}d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a^{2} c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a \,b^{2}+2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a^{3} \] Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**7,x)*b**2*c + 2*int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**6,x)*a*b*c + int(sqrt(sec(c + d*x)*b + a)*sec( c + d*x)**5,x)*a**2*c + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**5,x)*a* b**2 + 2*int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**4,x)*a**2*b + int(sqrt (sec(c + d*x)*b + a)*sec(c + d*x)**3,x)*a**3