Integrand size = 43, antiderivative size = 441 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d}+\frac {2 (9 b B+8 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d} \]
2/315*b*(261*B*a^2*b+75*B*b^3+64*a^3*C+2*a*b^2*(147*A+101*C))*sec(d*x+c)^( 3/2)*sin(d*x+c)/d+2/315*(1098*B*a^3*b+756*B*a*b^3+192*a^4*C+21*b^4*(9*A+7* C)+7*a^2*b^2*(261*A+155*C))*sin(d*x+c)*sec(d*x+c)^(1/2)/d+2/315*(63*A*b^2+ 117*B*a*b+48*C*a^2+49*C*b^2)*(a+b*sec(d*x+c))^2*sin(d*x+c)*sec(d*x+c)^(1/2 )/d+2/63*(9*B*b+8*C*a)*(a+b*sec(d*x+c))^3*sin(d*x+c)*sec(d*x+c)^(1/2)/d+2/ 9*C*(a+b*sec(d*x+c))^4*sin(d*x+c)*sec(d*x+c)^(1/2)/d-2/15*(60*B*a^3*b+36*B *a*b^3-15*a^4*(A-C)+18*a^2*b^2*(5*A+3*C)+b^4*(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+2/21*(21*B*a^4+42*B*a^2*b^2+5*B*b^4+28*a^3*b *(3*A+C)+4*a*b^3*(7*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c )*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/ d
Time = 12.30 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.38 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (105 a^4 A-630 a^2 A b^2-63 A b^4-420 a^3 b B-252 a b^3 B-105 a^4 C-378 a^2 b^2 C-49 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (420 a^3 A b+140 a A b^3+105 a^4 B+210 a^2 b^2 B+25 b^4 B+140 a^3 b C+100 a b^3 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{105 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} \left (90 a^2 A b^2+9 A b^4+60 a^3 b B+36 a b^3 B+15 a^4 C+54 a^2 b^2 C+7 b^4 C\right ) \sin (c+d x)+\frac {4}{7} \sec ^3(c+d x) \left (b^4 B \sin (c+d x)+4 a b^3 C \sin (c+d x)\right )+\frac {4}{21} \sec (c+d x) \left (28 a A b^3 \sin (c+d x)+42 a^2 b^2 B \sin (c+d x)+5 b^4 B \sin (c+d x)+28 a^3 b C \sin (c+d x)+20 a b^3 C \sin (c+d x)\right )+\frac {4}{45} \sec ^2(c+d x) \left (9 A b^4 \sin (c+d x)+36 a b^3 B \sin (c+d x)+54 a^2 b^2 C \sin (c+d x)+7 b^4 C \sin (c+d x)\right )+\frac {4}{9} b^4 C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sqrt[Sec[c + d*x]],x]
(2*Cos[c + d*x]^6*((2*(105*a^4*A - 630*a^2*A*b^2 - 63*A*b^4 - 420*a^3*b*B - 252*a*b^3*B - 105*a^4*C - 378*a^2*b^2*C - 49*b^4*C)*EllipticE[(c + d*x)/ 2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(420*a^3*A*b + 140*a*A* b^3 + 105*a^4*B + 210*a^2*b^2*B + 25*b^4*B + 140*a^3*b*C + 100*a*b^3*C)*Sq rt[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))/(105*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*(90*a^2*A*b^2 + 9* A*b^4 + 60*a^3*b*B + 36*a*b^3*B + 15*a^4*C + 54*a^2*b^2*C + 7*b^4*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^3*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d* x]))/7 + (4*Sec[c + d*x]*(28*a*A*b^3*Sin[c + d*x] + 42*a^2*b^2*B*Sin[c + d *x] + 5*b^4*B*Sin[c + d*x] + 28*a^3*b*C*Sin[c + d*x] + 20*a*b^3*C*Sin[c + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^4*Sin[c + d*x] + 36*a*b^3*B*Sin[c + d *x] + 54*a^2*b^2*C*Sin[c + d*x] + 7*b^4*C*Sin[c + d*x]))/45 + (4*b^4*C*Sec [c + d*x]^3*Tan[c + d*x])/9))/(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 = 3.10 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.02, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4584, 27, 3042, 4584, 27, 3042, 4584, 27, 3042, 4564, 27, 3042, 4535, 3042, 4258, 3042, 3120, 4534, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \sec (c+d x))^3 \left ((9 b B+8 a C) \sec ^2(c+d x)+(9 A b+7 C b+9 a B) \sec (c+d x)+a (9 A-C)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \sec (c+d x))^3 \left ((9 b B+8 a C) \sec ^2(c+d x)+(9 A b+7 C b+9 a B) \sec (c+d x)+a (9 A-C)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((9 b B+8 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 A b+7 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (9 A-C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \sec (c+d x))^2 \left (\left (48 C a^2+117 b B a+63 A b^2+49 b^2 C\right ) \sec ^2(c+d x)+\left (63 B a^2+126 A b a+82 b C a+45 b^2 B\right ) \sec (c+d x)+3 a (21 a A-3 b B-5 a C)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \sec (c+d x))^2 \left (\left (48 C a^2+117 b B a+63 A b^2+49 b^2 C\right ) \sec ^2(c+d x)+\left (63 B a^2+126 A b a+82 b C a+45 b^2 B\right ) \sec (c+d x)+3 a (21 a A-3 b B-5 a C)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (\left (48 C a^2+117 b B a+63 A b^2+49 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (63 B a^2+126 A b a+82 b C a+45 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (21 a A-3 b B-5 a C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int -\frac {(a+b \sec (c+d x)) \left (-3 \left (64 C a^3+261 b B a^2+2 b^2 (147 A+101 C) a+75 b^3 B\right ) \sec ^2(c+d x)-\left (315 B a^3+b (945 A+479 C) a^2+531 b^2 B a+21 b^3 (9 A+7 C)\right ) \sec (c+d x)+a \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right )\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}-\frac {1}{5} \int \frac {(a+b \sec (c+d x)) \left (-3 \left (64 C a^3+261 b B a^2+2 b^2 (147 A+101 C) a+75 b^3 B\right ) \sec ^2(c+d x)-\left (315 B a^3+b (945 A+479 C) a^2+531 b^2 B a+21 b^3 (9 A+7 C)\right ) \sec (c+d x)+a \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right )\right )}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}-\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-3 \left (64 C a^3+261 b B a^2+2 b^2 (147 A+101 C) a+75 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (-315 B a^3-b (945 A+479 C) a^2-531 b^2 B a-21 b^3 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right )\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}-\frac {2}{3} \int \frac {3 \left (\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2-\left (192 C a^4+1098 b B a^3+7 b^2 (261 A+155 C) a^2+756 b^3 B a+21 b^4 (9 A+7 C)\right ) \sec ^2(c+d x)-15 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right ) \sec (c+d x)\right )}{2 \sqrt {\sec (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2-\left (192 C a^4+1098 b B a^3+7 b^2 (261 A+155 C) a^2+756 b^3 B a+21 b^4 (9 A+7 C)\right ) \sec ^2(c+d x)-15 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2+\left (-192 C a^4-1098 b B a^3-7 b^2 (261 A+155 C) a^2-756 b^3 B a-21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-15 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2+\left (-192 C a^4-1098 b B a^3-7 b^2 (261 A+155 C) a^2-756 b^3 B a-21 b^4 (9 A+7 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}}dx+15 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \sqrt {\sec (c+d x)}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2+\left (-192 C a^4-1098 b B a^3-7 b^2 (261 A+155 C) a^2-756 b^3 B a-21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2+\left (-192 C a^4-1098 b B a^3-7 b^2 (261 A+155 C) a^2-756 b^3 B a-21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2+\left (-192 C a^4-1098 b B a^3-7 b^2 (261 A+155 C) a^2-756 b^3 B a-21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-\int \frac {\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) a^2+\left (-192 C a^4-1098 b B a^3-7 b^2 (261 A+155 C) a^2-756 b^3 B a-21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-21 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-21 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (-21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^2}{5 d}+\frac {1}{5} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}-\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}\) |
(2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d) + ((2*( 9*b*B + 8*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7* d) + ((2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*Sqrt[Sec[c + d*x]]*( a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((-42*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (30*(21*a^4*B + 4 2*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos [c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(1098*a^3* b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*(9*A + 7*C) + 7*a^2*b^2*(261*A + 15 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d)/5) /7)/9
3.11.5.3.1 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*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. \(2(461)=922\).
Time = 9.78 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.45
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1523\) |
| parts | \(\text {Expression too large to display}\) | \(1742\) |
int((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),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*B*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*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))+8*A*a^3*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c )^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic F(cos(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))-EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2)))+2*C*b^4*(-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+si n(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*b^3*(B*b+4*C*a)*(-1/56*cos(1/2*d*x+1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {15 \, \sqrt {2} {\left (21 i \, B a^{4} + 28 i \, {\left (3 \, A + C\right )} a^{3} b + 42 i \, B a^{2} b^{2} + 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-21 i \, B a^{4} - 28 i \, {\left (3 \, A + C\right )} a^{3} b - 42 i \, B a^{2} b^{2} - 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, {\left (A - C\right )} a^{4} + 60 i \, B a^{3} b + 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 i \, B a b^{3} + i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\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 ) + 21 \, \sqrt {2} {\left (15 i \, {\left (A - C\right )} a^{4} - 60 i \, B a^{3} b - 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} - 36 i \, B a b^{3} - i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\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 (35 \, C b^{4} + 21 \, {\left (15 \, C a^{4} + 60 \, B a^{3} b + 18 \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (28 \, C a^{3} b + 42 \, B a^{2} b^{2} + 4 \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (54 \, C a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\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 )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /2),x, algorithm="fricas")
-1/315*(15*sqrt(2)*(21*I*B*a^4 + 28*I*(3*A + C)*a^3*b + 42*I*B*a^2*b^2 + 4 *I*(7*A + 5*C)*a*b^3 + 5*I*B*b^4)*cos(d*x + c)^4*weierstrassPInverse(-4, 0 , cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-21*I*B*a^4 - 28*I*(3*A + C )*a^3*b - 42*I*B*a^2*b^2 - 4*I*(7*A + 5*C)*a*b^3 - 5*I*B*b^4)*cos(d*x + c) ^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)* (-15*I*(A - C)*a^4 + 60*I*B*a^3*b + 18*I*(5*A + 3*C)*a^2*b^2 + 36*I*B*a*b^ 3 + I*(9*A + 7*C)*b^4)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(15*I*(A - C)*a ^4 - 60*I*B*a^3*b - 18*I*(5*A + 3*C)*a^2*b^2 - 36*I*B*a*b^3 - I*(9*A + 7*C )*b^4)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co s(d*x + c) - I*sin(d*x + c))) - 2*(35*C*b^4 + 21*(15*C*a^4 + 60*B*a^3*b + 18*(5*A + 3*C)*a^2*b^2 + 36*B*a*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c)^4 + 15 *(28*C*a^3*b + 42*B*a^2*b^2 + 4*(7*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c)^ 3 + 7*(54*C*a^2*b^2 + 36*B*a*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c)^2 + 45*(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)^4)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /2),x, algorithm="maxima")
\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /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. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
int(((a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(1/2),x)