Integrand size = 43, antiderivative size = 515 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d} \]
2/693*(682*B*a^3*b+660*B*a*b^3+64*a^4*C+15*b^4*(11*A+9*C)+9*a^2*b^2*(143*A +101*C))*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/3465*b*(1353*B*a^2*b+539*B*b^3+19 2*a^3*C+2*a*b^2*(891*A+673*C))*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/231*(33*A*b ^2+55*B*a*b+16*C*a^2+27*C*b^2)*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^2*sin(d*x +c)/d+2/99*(11*B*b+8*C*a)*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^3*sin(d*x+c)/d +2/11*C*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^4*sin(d*x+c)/d+2/15*(15*B*a^4+54 *B*a^2*b^2+7*B*b^4+12*a^3*b*(5*A+3*C)+4*a*b^3*(9*A+7*C))*sin(d*x+c)*sec(d* x+c)^(1/2)/d-2/15*(15*B*a^4+54*B*a^2*b^2+7*B*b^4+12*a^3*b*(5*A+3*C)+4*a*b^ 3*(9*A+7*C))*(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)/d+2/231*(308*B* a^3*b+220*B*a*b^3+77*a^4*(3*A+C)+66*a^2*b^2*(7*A+5*C)+5*b^4*(11*A+9*C))*(c os(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)/d
Time = 13.20 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.38 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (-4620 a^3 A b-2772 a A b^3-1155 a^4 B-4158 a^2 b^2 B-539 b^4 B-2772 a^3 b C-2156 a b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (1155 a^4 A+2310 a^2 A b^2+275 A b^4+1540 a^3 b B+1100 a b^3 B+385 a^4 C+1650 a^2 b^2 C+225 b^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{1155 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} \left (60 a^3 A b+36 a A b^3+15 a^4 B+54 a^2 b^2 B+7 b^4 B+36 a^3 b C+28 a b^3 C\right ) \sin (c+d x)+\frac {4}{9} \sec ^4(c+d x) \left (b^4 B \sin (c+d x)+4 a b^3 C \sin (c+d x)\right )+\frac {4}{45} \sec ^2(c+d x) \left (36 a A b^3 \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+7 b^4 B \sin (c+d x)+36 a^3 b C \sin (c+d x)+28 a b^3 C \sin (c+d x)\right )+\frac {4}{77} \sec ^3(c+d x) \left (11 A b^4 \sin (c+d x)+44 a b^3 B \sin (c+d x)+66 a^2 b^2 C \sin (c+d x)+9 b^4 C \sin (c+d x)\right )+\frac {4}{231} \sec (c+d x) \left (462 a^2 A b^2 \sin (c+d x)+55 A b^4 \sin (c+d x)+308 a^3 b B \sin (c+d x)+220 a b^3 B \sin (c+d x)+77 a^4 C \sin (c+d x)+330 a^2 b^2 C \sin (c+d x)+45 b^4 C \sin (c+d x)\right )+\frac {4}{11} b^4 C \sec ^4(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \]
Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(2*Cos[c + d*x]^6*((2*(-4620*a^3*A*b - 2772*a*A*b^3 - 1155*a^4*B - 4158*a^ 2*b^2*B - 539*b^4*B - 2772*a^3*b*C - 2156*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(1155*a^4*A + 2310*a^2*A*b ^2 + 275*A*b^4 + 1540*a^3*b*B + 1100*a*b^3*B + 385*a^4*C + 1650*a^2*b^2*C + 225*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x ]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(1155* d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) ) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(6 0*a^3*A*b + 36*a*A*b^3 + 15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 36*a^3*b*C + 28*a*b^3*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^4*(b^4*B*Sin[c + d*x] + 4*a *b^3*C*Sin[c + d*x]))/9 + (4*Sec[c + d*x]^2*(36*a*A*b^3*Sin[c + d*x] + 54* a^2*b^2*B*Sin[c + d*x] + 7*b^4*B*Sin[c + d*x] + 36*a^3*b*C*Sin[c + d*x] + 28*a*b^3*C*Sin[c + d*x]))/45 + (4*Sec[c + d*x]^3*(11*A*b^4*Sin[c + d*x] + 44*a*b^3*B*Sin[c + d*x] + 66*a^2*b^2*C*Sin[c + d*x] + 9*b^4*C*Sin[c + d*x] ))/77 + (4*Sec[c + d*x]*(462*a^2*A*b^2*Sin[c + d*x] + 55*A*b^4*Sin[c + d*x ] + 308*a^3*b*B*Sin[c + d*x] + 220*a*b^3*B*Sin[c + d*x] + 77*a^4*C*Sin[c + d*x] + 330*a^2*b^2*C*Sin[c + d*x] + 45*b^4*C*Sin[c + d*x]))/231 + (4*b^4* C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2 *B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))
Time = 3.44 (sec) , antiderivative size = 482, normalized size of antiderivative = 0.94, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 4584, 27, 3042, 4584, 27, 3042, 4584, 27, 3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3119, 4534, 3042, 4258, 3042, 3120}
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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \left ((11 b B+8 a C) \sec ^2(c+d x)+(11 A b+9 C b+11 a B) \sec (c+d x)+a (11 A+C)\right )dx+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \left ((11 b B+8 a C) \sec ^2(c+d x)+(11 A b+9 C b+11 a B) \sec (c+d x)+a (11 A+C)\right )dx+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((11 b B+8 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(11 A b+9 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (11 A+C)\right )dx+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \sec ^2(c+d x)+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \sec (c+d x)+a (99 a A+11 b B+17 a C)\right )dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \sec ^2(c+d x)+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \sec (c+d x)+a (99 a A+11 b B+17 a C)\right )dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (99 a A+11 b B+17 a C)\right )dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \sec ^2(c+d x)+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \sec (c+d x)+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )dx+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \sec ^2(c+d x)+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \sec (c+d x)+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )dx+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )dx+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\sec (c+d x)} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \sec (c+d x)\right )dx+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sec (c+d x)} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \sec (c+d x)\right )dx+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx+\int \sqrt {\sec (c+d x)} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)\right )dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \sec (c+d x))^2}{7 d}+\frac {1}{7} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{d}+231 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}\) |
(2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d) + ((2* (11*b*B + 8*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/( 9*d) + ((6*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sec[c + d*x]^(3/2)* (a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2*b*(1353*a^2*b*B + 539*b^3 *B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x]) /(5*d) + ((30*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*( 7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15 *b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d + 231*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqr t[Sec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/5)/7)/9)/11
3.11.4.3.1 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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
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_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
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_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[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)*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]
Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(531)=1062\).
Time = 11.84 (sec) , antiderivative size = 1523, normalized size of antiderivative = 2.96
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1523\) |
| parts | \(\text {Expression too large to display}\) | \(1905\) |
int(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*a^4*(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))+ 2*C*b^4*(-1/352*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*si n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^ 4-15/154*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(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)))+2*b^3*(B*b+4*C*a)*(-1/144 *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)/( cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2 *d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x +1/2*c)^2)^(1/2)+7/15*(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))-7/15*(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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 2))))+2*a^3*(4*A*b+B*a)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.14 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {15 \, \sqrt {2} {\left (77 i \, {\left (3 \, A + C\right )} a^{4} + 308 i \, B a^{3} b + 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 i \, B a b^{3} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-77 i \, {\left (3 \, A + C\right )} a^{4} - 308 i \, B a^{3} b - 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} - 220 i \, B a b^{3} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (15 i \, B a^{4} + 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (-15 i \, B a^{4} - 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} - 7 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (231 \, {\left (15 \, B a^{4} + 12 \, {\left (5 \, A + 3 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 4 \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 315 \, C b^{4} + 15 \, {\left (77 \, C a^{4} + 308 \, B a^{3} b + 66 \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 \, B a b^{3} + 5 \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 77 \, {\left (36 \, C a^{3} b + 54 \, B a^{2} b^{2} + 4 \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left (66 \, C a^{2} b^{2} + 44 \, B a b^{3} + {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 385 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d \cos \left (d x + c\right )^{5}} \]
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
-1/3465*(15*sqrt(2)*(77*I*(3*A + C)*a^4 + 308*I*B*a^3*b + 66*I*(7*A + 5*C) *a^2*b^2 + 220*I*B*a*b^3 + 5*I*(11*A + 9*C)*b^4)*cos(d*x + c)^5*weierstras sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-77*I*(3*A + C)*a^4 - 308*I*B*a^3*b - 66*I*(7*A + 5*C)*a^2*b^2 - 220*I*B*a*b^3 - 5*I*( 11*A + 9*C)*b^4)*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(15*I*B*a^4 + 12*I*(5*A + 3*C)*a^3*b + 54*I* B*a^2*b^2 + 4*I*(9*A + 7*C)*a*b^3 + 7*I*B*b^4)*cos(d*x + c)^5*weierstrassZ eta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 23 1*sqrt(2)*(-15*I*B*a^4 - 12*I*(5*A + 3*C)*a^3*b - 54*I*B*a^2*b^2 - 4*I*(9* A + 7*C)*a*b^3 - 7*I*B*b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstr assPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(231*(15*B*a^4 + 12 *(5*A + 3*C)*a^3*b + 54*B*a^2*b^2 + 4*(9*A + 7*C)*a*b^3 + 7*B*b^4)*cos(d*x + c)^5 + 315*C*b^4 + 15*(77*C*a^4 + 308*B*a^3*b + 66*(7*A + 5*C)*a^2*b^2 + 220*B*a*b^3 + 5*(11*A + 9*C)*b^4)*cos(d*x + c)^4 + 77*(36*C*a^3*b + 54*B *a^2*b^2 + 4*(9*A + 7*C)*a*b^3 + 7*B*b^4)*cos(d*x + c)^3 + 45*(66*C*a^2*b^ 2 + 44*B*a*b^3 + (11*A + 9*C)*b^4)*cos(d*x + c)^2 + 385*(4*C*a*b^3 + B*b^4 )*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^5)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \]
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^4*s qrt(sec(d*x + c)), x)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]