Integrand size = 35, antiderivative size = 286 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {4 a^3 (17 A+27 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (11 A+21 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (17 A+27 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (16 A+21 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (73 A+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{21 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \] Output:
-4/15*a^3*(17*A+27*C)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2 ))*sec(d*x+c)^(1/2)/d+4/21*a^3*(11*A+21*C)*cos(d*x+c)^(1/2)*InverseJacobiA M(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+4/15*a^3*(17*A+27*C)*sec(d*x+c )^(1/2)*sin(d*x+c)/d+8/105*a^3*(16*A+21*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2 /315*(73*A+63*C)*(a^3+a^3*cos(d*x+c))*sec(d*x+c)^(5/2)*sin(d*x+c)/d+4/21*A *(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d+2/9*A*(a+a*cos(d*x +c))^3*sec(d*x+c)^(9/2)*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.61 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.29 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {17 A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^3 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{90 \sqrt {2} d}+\frac {3 C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^3 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{10 \sqrt {2} d}+\frac {11 A \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{42 d}+\frac {C \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{2 d}+(a+a \cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (\frac {(17 A+27 C) \cos (d x) \csc (c)}{30 d}+\frac {A \sec (c) \sec ^4(c+d x) \sin (d x)}{36 d}+\frac {\sec (c) \sec ^3(c+d x) (7 A \sin (c)+27 A \sin (d x))}{252 d}+\frac {\sec (c) \sec ^2(c+d x) (135 A \sin (c)+238 A \sin (d x)+63 C \sin (d x))}{1260 d}+\frac {\sec (c) \sec (c+d x) (238 A \sin (c)+63 C \sin (c)+330 A \sin (d x)+315 C \sin (d x))}{1260 d}+\frac {(22 A+21 C) \tan (c)}{84 d}\right ) \] Input:
Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2 ),x]
Output:
(17*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2 *I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(90*Sqrt[2]*d*E^(I*d*x)) + (3*C*Sqr t[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))] *(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I )*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(10*Sqrt[2]*d*E^(I*d*x)) + (11*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2 ]^6*Sqrt[Sec[c + d*x]])/(42*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x] )^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]])/(2* d) + (a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*(((17* A + 27*C)*Cos[d*x]*Csc[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(36 *d) + (Sec[c]*Sec[c + d*x]^3*(7*A*Sin[c] + 27*A*Sin[d*x]))/(252*d) + (Sec[ c]*Sec[c + d*x]^2*(135*A*Sin[c] + 238*A*Sin[d*x] + 63*C*Sin[d*x]))/(1260*d ) + (Sec[c]*Sec[c + d*x]*(238*A*Sin[c] + 63*C*Sin[c] + 330*A*Sin[d*x] + 31 5*C*Sin[d*x]))/(1260*d) + ((22*A + 21*C)*Tan[c])/(84*d))
Time = 1.92 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.93, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.657, Rules used = {3042, 4709, 3042, 3523, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3447, 3042, 3500, 27, 3042, 3227, 3042, 3116, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{11/2} (a \cos (c+d x)+a)^3 \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(\cos (c+d x) a+a)^3 \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {(\cos (c+d x) a+a)^3 (6 a A+a (A+9 C) \cos (c+d x))}{2 \cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(\cos (c+d x) a+a)^3 (6 a A+a (A+9 C) \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (6 a A+a (A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((73 A+63 C) a^2+(13 A+63 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((73 A+63 C) a^2+(13 A+63 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((73 A+63 C) a^2+(13 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left (6 (16 A+21 C) a^3+(23 A+63 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(\cos (c+d x) a+a) \left (6 (16 A+21 C) a^3+(23 A+63 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (6 (16 A+21 C) a^3+(23 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(23 A+63 C) \cos ^2(c+d x) a^4+6 (16 A+21 C) a^4+\left (6 (16 A+21 C) a^4+(23 A+63 C) a^4\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(23 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+6 (16 A+21 C) a^4+\left (6 (16 A+21 C) a^4+(23 A+63 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {2}{3} \int \frac {3 \left (7 (17 A+27 C) a^4+5 (11 A+21 C) \cos (c+d x) a^4\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {7 (17 A+27 C) a^4+5 (11 A+21 C) \cos (c+d x) a^4}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {7 (17 A+27 C) a^4+5 (11 A+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (17 A+27 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 a^4 (11 A+21 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (17 A+27 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 a^4 (11 A+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (17 A+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (17 A+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+7 a^4 (17 A+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{7} \left (\frac {2 (73 A+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {6}{5} \left (\frac {10 a^4 (11 A+21 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {4 a^4 (16 A+21 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+7 a^4 (17 A+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {12 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )\) |
Input:
Int[(a + a*Cos[c + d*x])^3*(A + 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*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((12*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*(73*A + 63*C)*(a^4 + a^4*Cos[c + d* x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (6*((10*a^4*(11*A + 21*C)*Ell ipticF[(c + d*x)/2, 2])/d + (4*a^4*(16*A + 21*C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + 7*a^4*(17*A + 27*C)*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*S in[c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/(9*a))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 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. \(1218\) vs. \(2(257)=514\).
Time = 1121.98 (sec) , antiderivative size = 1219, normalized size of antiderivative = 4.26
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1219\) |
| parts | \(\text {Expression too large to display}\) | \(1548\) |
Input:
int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x,method=_RETU RNVERBOSE)
Output:
-16*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(1/8*C*( 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))+1/8*A*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x +1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2 *sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/ 2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c )^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*c os(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2 )^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d *x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/ 2*d*x+1/2*c),2^(1/2))))+3/8*A*(-1/56*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)^4-5/42*cos(1 /2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/ 2*d*x+1/2*c)^2-1/2)^2+5/21*(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)*Ellip ticF(cos(1/2*d*x+1/2*c),2^(1/2)))+3/8*C/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d* x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin( 1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipt...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.98 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (11 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (11 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (17 \, A + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (17 \, A + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (17 \, A + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (22 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (34 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 135 \, A a^{3} \cos \left (d x + c\right ) + 35 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, algor ithm="fricas")
Output:
-2/315*(15*I*sqrt(2)*(11*A + 21*C)*a^3*cos(d*x + c)^4*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(11*A + 21*C)*a^3*cos (d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21 *I*sqrt(2)*(17*A + 27*C)*a^3*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierst rassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*(17*A + 27*C)*a^3*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , cos(d*x + c) - I*sin(d*x + c))) - (42*(17*A + 27*C)*a^3*cos(d*x + c)^4 + 15*(22*A + 21*C)*a^3*cos(d*x + c)^3 + 7*(34*A + 9*C)*a^3*cos(d*x + c)^2 + 135*A*a^3*cos(d*x + c) + 35*A*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*co s(d*x + c)^4)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(11/2),x)
Output:
Timed out
\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, algor ithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(11/2 ), x)
\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x, algor ithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(11/2 ), x)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(11/2)*(a + a*cos(c + d*x))^3, x)
Output:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(11/2)*(a + a*cos(c + d*x))^3, x)
\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=a^{3} \left (3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{5}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{5}d x \right ) a \right ) \] Input:
int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2),x)
Output:
a**3*(3*int(sqrt(sec(c + d*x))*cos(c + d*x)*sec(c + d*x)**5,x)*a + int(sqr t(sec(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**5,x)*c + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**5,x)*c + int(sqrt(sec(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**5,x)*a + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**3* sec(c + d*x)**5,x)*c + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d* x)**5,x)*a + int(sqrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d*x)**5,x)*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**5,x)*a)