Integrand size = 45, antiderivative size = 455 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}-\frac {2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d} \] Output:
2/315*(a^2-b^2)*(8*A*b^3+75*B*a^3-18*B*a*b^2+a^2*(39*A*b+63*C*b))*((b+a*co s(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/ 2))/a^3/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+2/315*(8*A*b^4+246*B*a^3 *b-18*B*a*b^3+21*a^4*(7*A+9*C)+3*a^2*b^2*(11*A+21*C))*cos(d*x+c)^(1/2)*Ell ipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/ a^3/d/((b+a*cos(d*x+c))/(a+b))^(1/2)-2/315*(4*A*b^3-75*B*a^3-9*B*a*b^2-2*a ^2*b*(44*A+63*C))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^2/d +2/315*(3*A*b^2+72*B*a*b+7*a^2*(7*A+9*C))*cos(d*x+c)^(3/2)*(a+b*sec(d*x+c) )^(1/2)*sin(d*x+c)/a/d+2/21*(A*b+3*B*a)*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^ (1/2)*sin(d*x+c)/d+2/9*A*cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c )/d
Result contains complex when optimal does not.
Time = 31.87 (sec) , antiderivative size = 3703, normalized size of antiderivative = 8.14 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x ] + C*Sec[c + d*x]^2),x]
Output:
(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec [c + d*x]^2)*(((402*a^2*A*b - 16*A*b^3 + 345*a^3*B + 36*a*b^2*B + 504*a^2* b*C)*Sin[c + d*x])/(315*a^2) + ((133*a^2*A + 6*A*b^2 + 144*a*b*B + 126*a^2 *C)*Sin[2*(c + d*x)])/(315*a) + ((10*A*b + 9*a*B)*Sin[3*(c + d*x)])/63 + ( a*A*Sin[4*(c + d*x)])/18))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((14*a^2*A*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (22*A*b^2*Sqrt [Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A* b^4*Sqrt[Cos[c + d*x]])/(315*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x ]]) + (164*a*b*B*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Se c[c + d*x]]) - (4*b^3*B*Sqrt[Cos[c + d*x]])/(35*a*Sqrt[b + a*Cos[c + d*x]] *Sqrt[Sec[c + d*x]]) + (6*a^2*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos [c + d*x]]*Sqrt[Sec[c + d*x]]) + (124*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt [Sec[c + d*x]])/(315*a*Sqrt[b + a*Cos[c + d*x]]) + (10*a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (34*b^2*B*Sqrt[C os[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (8*a*b*C* Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]))*(Cos[ (c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Se...
Time = 4.25 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.09, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4753, 3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
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.
| \(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{9/2} (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \sec (c+d x))^{3/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (4 A+9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+3 (A b+3 a B)\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (4 A+9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+3 (A b+3 a B)\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (4 A+9 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 (A b+3 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {7 (7 A+9 C) a^2+72 b B a+3 A b^2+b (40 A b+63 C b+36 a B) \sec ^2(c+d x)+\left (45 B a^2+92 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {7 (7 A+9 C) a^2+72 b B a+3 A b^2+b (40 A b+63 C b+36 a B) \sec ^2(c+d x)+\left (45 B a^2+92 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {7 (7 A+9 C) a^2+72 b B a+3 A b^2+b (40 A b+63 C b+36 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+92 A b a+126 b C a+63 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left (7 (7 A+9 C) a^2+72 b B a+3 A b^2\right ) \sec ^2(c+d x)-a \left (21 (7 A+9 C) a^2+396 b B a+b^2 (209 A+315 C)\right ) \sec (c+d x)+3 \left (-75 B a^3-2 b (44 A+63 C) a^2-9 b^2 B a+4 A b^3\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (7 (7 A+9 C) a^2+72 b B a+3 A b^2\right ) \sec ^2(c+d x)-a \left (21 (7 A+9 C) a^2+396 b B a+b^2 (209 A+315 C)\right ) \sec (c+d x)+3 \left (-75 B a^3-2 b (44 A+63 C) a^2-9 b^2 B a+4 A b^3\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (7 (7 A+9 C) a^2+72 b B a+3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (21 (7 A+9 C) a^2+396 b B a+b^2 (209 A+315 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-75 B a^3-2 b (44 A+63 C) a^2-9 b^2 B a+4 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+\left (75 B a^3+6 b (31 A+42 C) a^2+153 b^2 B a+2 A b^3\right ) \sec (c+d x) a+8 A b^4\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+\left (75 B a^3+6 b (31 A+42 C) a^2+153 b^2 B a+2 A b^3\right ) \sec (c+d x) a+8 A b^4}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+\left (75 B a^3+6 b (31 A+42 C) a^2+153 b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+8 A b^4}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 (3 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
Input:
Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C* Sec[c + d*x]^2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + b*Sec[c + d*x])^(3/2)*Sin [c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((6*(A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d *x]^(3/2)) - (-(((2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39 *A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8* A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a* d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a) + (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Sec[c + d*x]]* Sin[c + d*x])/(a*d*Sqrt[Sec[c + d*x]]))/(5*a))/7)/9)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && 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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*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] && LeQ[n, -1]
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Leaf count of result is larger than twice the leaf count of optimal. \(3632\) vs. \(2(424)=848\).
Time = 32.88 (sec) , antiderivative size = 3633, normalized size of antiderivative = 7.98
Input:
int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x,method=_RETURNVERBOSE)
Output:
2/315/d*((-246*cos(d*x+c)^2-492*cos(d*x+c)-246)*B*(1/(a+b)*(b+a*cos(d*x+c) )/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticE(((a-b)/ (a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-18*cos(d*x+c) ^2-36*cos(d*x+c)-18)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/ (cos(d*x+c)+1))^(1/2)*a^2*b^3*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-co t(d*x+c)),(-(a+b)/(a-b))^(1/2))+(18*cos(d*x+c)^2+36*cos(d*x+c)+18)*B*(1/(a +b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a*b^4* EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2) )+(63*cos(d*x+c)^2+126*cos(d*x+c)+63)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x +c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticE(((a-b)/(a+b))^(1/ 2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-63*cos(d*x+c)^2-126*cos (d*x+c)-63)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+ c)+1))^(1/2)*a^2*b^3*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)) ,(-(a+b)/(a-b))^(1/2))+(186*cos(d*x+c)^2+372*cos(d*x+c)+186)*A*(1/(a+b)*(b +a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b*Ellipt icF(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-33 *cos(d*x+c)^2-66*cos(d*x+c)-33)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1) )^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*EllipticF(((a-b)/(a+b))^(1/2)*(cs c(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(2*cos(d*x+c)^2+4*cos(d*x+c)+2) *A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.58 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="fricas")
Output:
2/945*(3*(35*A*a^5*cos(d*x + c)^3 + 75*B*a^5 + 2*(44*A + 63*C)*a^4*b + 9*B *a^3*b^2 - 4*A*a^2*b^3 + 5*(9*B*a^5 + 10*A*a^4*b)*cos(d*x + c)^2 + (7*(7*A + 9*C)*a^5 + 72*B*a^4*b + 3*A*a^3*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - sqrt(1/2)*(225*I*B*a ^5 + 6*I*(44*A + 63*C)*a^4*b - 33*I*B*a^3*b^2 - 6*I*(10*A + 21*C)*a^2*b^3 + 36*I*B*a*b^4 - 16*I*A*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b ^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) - sqrt(1/2)*(-225*I*B*a^5 - 6*I*(44*A + 63*C)*a^4*b + 33*I *B*a^3*b^2 + 6*I*(10*A + 21*C)*a^2*b^3 - 36*I*B*a*b^4 + 16*I*A*b^5)*sqrt(a )*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3 , 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(1/2)*(-21* I*(7*A + 9*C)*a^5 - 246*I*B*a^4*b - 3*I*(11*A + 21*C)*a^3*b^2 + 18*I*B*a^2 *b^3 - 8*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27 *(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27 *(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/ a)) - 3*sqrt(1/2)*(21*I*(7*A + 9*C)*a^5 + 246*I*B*a^4*b + 3*I*(11*A + 21*C )*a^3*b^2 - 18*I*B*a^2*b^3 + 8*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3* a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3* a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I* a*sin(d*x + c) + 2*b)/a)))/(a^4*d)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec( d*x+c)**2),x)
Output:
Timed out
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/ 2)*cos(d*x + c)^(9/2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/ 2)*cos(d*x + c)^(9/2), x)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{9/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int(cos(c + d*x)^(9/2)*(a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)^(9/2)*(a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) b^{2}+2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d* x)**3,x)*b*c + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x )**4*sec(c + d*x)**2,x)*a*c + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d* x))*cos(c + d*x)**4*sec(c + d*x)**2,x)*b**2 + 2*int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x),x)*a*b + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*a**2