Integrand size = 45, antiderivative size = 592 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^3+12 a b^2 (A-2 B)+6 a^2 b (6 A-3 B+7 C)+3 a^3 (49 A-25 B+63 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \] Output:
-2/315*(a-b)*(a+b)^(1/2)*(16*A*b^4-57*B*a^3*b-24*B*a*b^3+6*a^2*b^2*(4*A+7* C)-21*a^4*(7*A+9*C))*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticE((a+b*cos(d*x+c) )^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c ))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^5/d/sec(d*x+c)^(1/2)-2/31 5*(a-b)*(a+b)^(1/2)*(16*A*b^3+12*a*b^2*(A-2*B)+6*a^2*b*(6*A-3*B+7*C)+3*a^3 *(49*A-25*B+63*C))*cos(d*x+c)^(1/2)*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^ (1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c)) /(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d/sec(d*x+c)^(1/2)+2/315* (8*A*b^3+75*B*a^3-12*B*a*b^2+a^2*b*(13*A+21*C))*(a+b*cos(d*x+c))^(1/2)*sec (d*x+c)^(3/2)*sin(d*x+c)/a^3/d-2/315*(6*A*b^2-9*B*a*b-7*a^2*(7*A+9*C))*(a+ b*cos(d*x+c))^(1/2)*sec(d*x+c)^(5/2)*sin(d*x+c)/a^2/d+2/63*(A*b+9*B*a)*(a+ b*cos(d*x+c))^(1/2)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d+2/9*A*(a+b*cos(d*x+c)) ^(1/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(4669\) vs. \(2(592)=1184\).
Time = 26.71 (sec) , antiderivative size = 4669, normalized size of antiderivative = 7.89 \[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Result too large to show} \] Input:
Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2) *Sec[c + d*x]^(11/2),x]
Output:
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B + 189*a^4*C - 42*a^2*b^2*C)*Sin[c + d*x])/(315*a^4) + (2*Sec[c + d*x]^3*(A*b*Sin[c + d*x] + 9*a*B*Sin[c + d*x] ))/(63*a) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] - 6*A*b^2*Sin[c + d*x ] + 9*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(13*a^2*A*b*Sin[c + d*x] + 8*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d* x] - 12*a*b^2*B*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/(315*a^3) + (2*A* Sec[c + d*x]^3*Tan[c + d*x])/9))/d + (2*((-7*a*A)/(15*Sqrt[a + b*Cos[c + d *x]]*Sqrt[Sec[c + d*x]]) + (8*A*b^2)/(105*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[ Sec[c + d*x]]) + (16*A*b^4)/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (19*b*B)/(105*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8* b^3*B)/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a*C)/(5* Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C)/(15*a*Sqrt[a + b* Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b*Sqrt[Sec[c + d*x]])/(35*Sqrt[a + b*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Sqrt[Sec[c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x] ]) + (5*a*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (17*b^2*B* Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^4*B*Sqrt[Sec[c + d*x]])/(105*a^3*Sqrt[a + b*Cos[c + d*x]]) - (2*b*C*Sqrt[Sec[c + d*x]])/ (15*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(15*a^2*Sq...
Time = 3.28 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.422, Rules used = {3042, 4709, 3042, 3526, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3477, 3042, 3295, 3473}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^{\frac {11}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{11/2} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {a+b \cos (c+d x)} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {3 b (2 A+3 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+A b+9 a B}{2 \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 A+3 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+A b+9 a B}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+A b+9 a B}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 \int -\frac {-7 (7 A+9 C) a^2-9 b B a-(47 A b+63 C b+45 a B) \cos (c+d x) a+6 A b^2-4 b (A b+9 a B) \cos ^2(c+d x)}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}+\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-7 (7 A+9 C) a^2-9 b B a-(47 A b+63 C b+45 a B) \cos (c+d x) a+6 A b^2-4 b (A b+9 a B) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\int \frac {-7 (7 A+9 C) a^2-9 b B a-(47 A b+63 C b+45 a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2-4 b (A b+9 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-2 b \left (-7 (7 A+9 C) a^2-9 b B a+6 A b^2\right ) \cos ^2(c+d x)+a \left (21 (7 A+9 C) a^2+207 b B a+2 A b^2\right ) \cos (c+d x)+3 \left (75 B a^3+b (13 A+21 C) a^2-12 b^2 B a+8 A b^3\right )}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (-7 (7 A+9 C) a^2-9 b B a+6 A b^2\right ) \cos ^2(c+d x)+a \left (21 (7 A+9 C) a^2+207 b B a+2 A b^2\right ) \cos (c+d x)+3 \left (75 B a^3+b (13 A+21 C) a^2-12 b^2 B a+8 A b^3\right )}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (-7 (7 A+9 C) a^2-9 b B a+6 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (21 (7 A+9 C) a^2+207 b B a+2 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (75 B a^3+b (13 A+21 C) a^2-12 b^2 B a+8 A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {3 \left (-21 (7 A+9 C) a^4-57 b B a^3+6 b^2 (4 A+7 C) a^2-24 b^3 B a+\left (-75 B a^3-3 b (37 A+49 C) a^2-6 b^2 B a+4 A b^3\right ) \cos (c+d x) a+16 A b^4\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-21 (7 A+9 C) a^4-57 b B a^3+6 b^2 (4 A+7 C) a^2-24 b^3 B a+\left (-75 B a^3-3 b (37 A+49 C) a^2-6 b^2 B a+4 A b^3\right ) \cos (c+d x) a+16 A b^4}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-21 (7 A+9 C) a^4-57 b B a^3+6 b^2 (4 A+7 C) a^2-24 b^3 B a+\left (-75 B a^3-3 b (37 A+49 C) a^2-6 b^2 B a+4 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+16 A b^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (3 a^3 (49 A-25 B+63 C)+6 a^2 b (6 A-3 B+7 C)+12 a b^2 (A-2 B)+16 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (3 a^3 (49 A-25 B+63 C)+6 a^2 b (6 A-3 B+7 C)+12 a b^2 (A-2 B)+16 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (3 a^3 (49 A-25 B+63 C)+6 a^2 b (6 A-3 B+7 C)+12 a b^2 (A-2 B)+16 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (3 a^3 (49 A-25 B+63 C)+6 a^2 b (6 A-3 B+7 C)+12 a b^2 (A-2 B)+16 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
Input:
Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*(A*b + 9*a*B)*Sqrt[a + b*Cos[c + d *x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - ((2*(6*A*b^2 - 9*a*b*B - 7 *a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d* x]^(5/2)) - (-(((2*(a - b)*Sqrt[a + b]*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cot[c + d*x]*EllipticE[ArcS in[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/( a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/ (a - b)])/(a^2*d) + (2*(a - b)*Sqrt[a + b]*(16*A*b^3 + 12*a*b^2*(A - 2*B) + 6*a^2*b*(6*A - 3*B + 7*C) + 3*a^3*(49*A - 25*B + 63*C))*Cot[c + d*x]*Ell ipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[ c + d*x]))/(a - b)])/(a*d))/a) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2 *b*(13*A + 21*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Cos[c + d*x] ^(3/2)))/(5*a))/(7*a))/9)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(3628\) vs. \(2(534)=1068\).
Time = 73.08 (sec) , antiderivative size = 3629, normalized size of antiderivative = 6.13
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3629\) |
| parts | \(\text {Expression too large to display}\) | \(3690\) |
Input:
int((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/ 2),x,method=_RETURNVERBOSE)
Output:
2/315/d/a^4*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^(11/2)/(b*cos(d*x+c)^2+a*cos (d*x+c)+b*cos(d*x+c)+a)*(C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2) *(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(- (a-b)/(a+b))^(1/2))*(-189*cos(d*x+c)^7-378*cos(d*x+c)^6-189*cos(d*x+c)^5)+ A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c )))^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(147* cos(d*x+c)^7+294*cos(d*x+c)^6+147*cos(d*x+c)^5)+A*(1/(a+b)*(a+b*cos(d*x+c) )/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^5*EllipticE(-c sc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-16*cos(d*x+c)^7-32*cos(d*x+c) ^6-16*cos(d*x+c)^5)+C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos (d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c),(-(a-b) /(a+b))^(1/2))*(189*cos(d*x+c)^7+378*cos(d*x+c)^6+189*cos(d*x+c)^5)+A*(1/( a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1 /2)*a^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-147*cos(d *x+c)^7-294*cos(d*x+c)^6-147*cos(d*x+c)^5)+B*(1/(a+b)*(a+b*cos(d*x+c))/(1+ cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticF(-csc(d* x+c)+cot(d*x+c),(-(a-b)/(a+b))^(1/2))*(-75*cos(d*x+c)^7-150*cos(d*x+c)^6-7 5*cos(d*x+c)^5)+sin(d*x+c)*cos(d*x+c)^3*(13*cos(d*x+c)^3-11*cos(d*x+c)^2-c os(d*x+c)-1)*A*a^3*b^2+sin(d*x+c)*cos(d*x+c)^4*(-24*cos(d*x+c)^2+2*cos(d*x +c)+2)*a^2*A*b^3+sin(d*x+c)*cos(d*x+c)^5*(8*cos(d*x+c)-8)*A*a*b^4+sin(d...
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(11/2),x, algorithm="fricas")
Output:
integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)* sec(d*x + c)^(11/2), x)
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**(11/2),x)
Output:
Timed out
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(11/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a) *sec(d*x + c)^(11/2), x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(11/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a) *sec(d*x + c)^(11/2), x)
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int((1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int((1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a \] Input:
int((a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/ 2),x)
Output:
int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)* *5,x)*b + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2* sec(c + d*x)**5,x)*c + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*sec (c + d*x)**5,x)*a