Integrand size = 35, antiderivative size = 319 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (105 A+143 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (5 A+7 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (105 A+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (35 A+44 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \] Output:
-4/5*a^3*(5*A+7*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+4/231*a^3*(105*A+143*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+4/5*a^3*(5*A+7*C)*sec(d*x+c)^( 1/2)*sin(d*x+c)/d+4/231*a^3*(105*A+143*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+8/ 385*a^3*(35*A+44*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/231*(35*A+33*C)*(a^3+a ^3*cos(d*x+c))*sec(d*x+c)^(7/2)*sin(d*x+c)/d+4/33*A*(a^2+a^2*cos(d*x+c))^2 *sec(d*x+c)^(9/2)*sin(d*x+c)/a/d+2/11*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^(11/ 2)*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.69 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.18 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^3 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{6 \sqrt {2} d}+\frac {7 C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^3 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{30 \sqrt {2} d}+\frac {5 A \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{22 d}+\frac {13 C \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{42 d}+(a+a \cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (\frac {(5 A+7 C) \cos (d x) \csc (c)}{10 d}+\frac {A \sec (c) \sec ^5(c+d x) \sin (d x)}{44 d}+\frac {\sec (c) \sec ^4(c+d x) (3 A \sin (c)+11 A \sin (d x))}{132 d}+\frac {\sec (c) \sec ^3(c+d x) (77 A \sin (c)+126 A \sin (d x)+33 C \sin (d x))}{924 d}+\frac {\sec (c) \sec ^2(c+d x) (630 A \sin (c)+165 C \sin (c)+770 A \sin (d x)+693 C \sin (d x))}{4620 d}+\frac {\sec (c) \sec (c+d x) (770 A \sin (c)+693 C \sin (c)+1050 A \sin (d x)+1430 C \sin (d x))}{4620 d}+\frac {(105 A+143 C) \tan (c)}{462 d}\right ) \] Input:
Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2 ),x]
Output:
(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)/(6*Sqrt[2]*d*E^(I*d*x)) + (7*C*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)/(30*Sqrt[2]*d*E^(I*d*x)) + (5*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*S qrt[Sec[c + d*x]])/(22*d) + (13*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]])/(42*d ) + (a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*(((5*A + 7*C)*Cos[d*x]*Csc[c])/(10*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^4*(3*A*Sin[c] + 11*A*Sin[d*x]))/(132*d) + (Sec[c]* Sec[c + d*x]^3*(77*A*Sin[c] + 126*A*Sin[d*x] + 33*C*Sin[d*x]))/(924*d) + ( Sec[c]*Sec[c + d*x]^2*(630*A*Sin[c] + 165*C*Sin[c] + 770*A*Sin[d*x] + 693* C*Sin[d*x]))/(4620*d) + (Sec[c]*Sec[c + d*x]*(770*A*Sin[c] + 693*C*Sin[c] + 1050*A*Sin[d*x] + 1430*C*Sin[d*x]))/(4620*d) + ((105*A + 143*C)*Tan[c])/ (462*d))
Time = 1.99 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \sec ^{\frac {13}{2}}(c+d x) (a \cos (c+d x)+a)^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{13/2} (a \cos (c+d x)+a)^3 \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(\cos (c+d x) a+a)^3 \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {(\cos (c+d x) a+a)^3 (6 a A+a (3 A+11 C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(\cos (c+d x) a+a)^3 (6 a A+a (3 A+11 C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (6 a A+a (3 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{9} \int \frac {3 (\cos (c+d x) a+a)^2 \left ((35 A+33 C) a^2+3 (5 A+11 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \frac {(\cos (c+d x) a+a)^2 \left ((35 A+33 C) a^2+3 (5 A+11 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((35 A+33 C) a^2+3 (5 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {2}{7} \int \frac {3 (\cos (c+d x) a+a) \left (2 (35 A+44 C) a^3+5 (7 A+11 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {(\cos (c+d x) a+a) \left (2 (35 A+44 C) a^3+5 (7 A+11 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (2 (35 A+44 C) a^3+5 (7 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {5 (7 A+11 C) \cos ^2(c+d x) a^4+2 (35 A+44 C) a^4+\left (5 (7 A+11 C) a^4+2 (35 A+44 C) a^4\right ) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \int \frac {5 (7 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+2 (35 A+44 C) a^4+\left (5 (7 A+11 C) a^4+2 (35 A+44 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {2}{5} \int \frac {5 (105 A+143 C) a^4+77 (5 A+7 C) \cos (c+d x) a^4}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {5 (105 A+143 C) a^4+77 (5 A+7 C) \cos (c+d x) a^4}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {5 (105 A+143 C) a^4+77 (5 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (105 A+143 C) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)}dx+77 a^4 (5 A+7 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (105 A+143 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+77 a^4 (5 A+7 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (105 A+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (5 A+7 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (105 A+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (5 A+7 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{5} \left (5 a^4 (105 A+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (5 A+7 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \left (\frac {2 (35 A+33 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {6}{7} \left (\frac {4 a^4 (35 A+44 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (5 a^4 (105 A+143 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (5 A+7 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {4 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
Input:
Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),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])/(11*d*Cos[c + d*x]^(11/2)) + ((4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(9/2)) + ((2*(35*A + 33*C)*(a^4 + a^4*Cos[c + d *x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*((4*a^4*(35*A + 44*C)*Sin [c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (5*a^4*(105*A + 143*C)*((2*EllipticF [(c + d*x)/2, 2])/(3*d) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))) + 77* a^4*(5*A + 7*C)*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sq rt[Cos[c + d*x]])))/5))/7)/3)/(11*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. \(1380\) vs. \(2(286)=572\).
Time = 1124.70 (sec) , antiderivative size = 1381, normalized size of antiderivative = 4.33
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1381\) |
| parts | \(\text {Expression too large to display}\) | \(1711\) |
Input:
int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),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*A*( -1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^( 1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*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)^4-15/154* cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(c os(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) *EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+3/8*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*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/ 2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^ (1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2) /(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d *x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+3/8*C*(-1/6*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+1/3*(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)*E...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.94 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (462 \, {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (10 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (42 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, A a^{3} \cos \left (d x + c\right ) + 105 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{1155 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algor ithm="fricas")
Output:
-2/1155*(5*I*sqrt(2)*(105*A + 143*C)*a^3*cos(d*x + c)^5*weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(2)*(105*A + 143*C)*a^3* cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*I*sqrt(2)*(5*A + 7*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*I*sqrt(2)*(5*A + 7*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (462*(5*A + 7*C)*a^3*cos(d*x + c)^5 + 10*(105*A + 143*C)*a^3*cos(d*x + c)^4 + 77*(10*A + 9*C)*a^3*cos(d*x + c)^ 3 + 15*(42*A + 11*C)*a^3*cos(d*x + c)^2 + 385*A*a^3*cos(d*x + c) + 105*A*a ^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^5)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(13/2),x)
Output:
Timed out
\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algor ithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(13/2 ), x)
\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algor ithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(13/2 ), x)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(13/2)*(a + a*cos(c + d*x))^3, x)
Output:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(13/2)*(a + a*cos(c + d*x))^3, x)
\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=a^{3} \left (3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{6}d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{6}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{6}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{6}d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{6}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{6}d x \right ) a \right ) \] Input:
int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x)
Output:
a**3*(3*int(sqrt(sec(c + d*x))*cos(c + d*x)*sec(c + d*x)**6,x)*a + int(sqr t(sec(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**6,x)*c + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**6,x)*c + int(sqrt(sec(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**6,x)*a + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**3* sec(c + d*x)**6,x)*c + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d* x)**6,x)*a + int(sqrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d*x)**6,x)*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**6,x)*a)