Integrand size = 37, antiderivative size = 627 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 (a-b) b \sqrt {a+b} \left (8 A b^4+3 a^2 b^2 (17 A+33 C)+a^4 (741 A+957 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (6 a A b^3+8 A b^4+15 a^4 (9 A+11 C)+3 a^2 b^2 (19 A+33 C)-6 a^3 b (101 A+132 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{693 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (4 A b^4-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{693 a^2 d}+\frac {2 b \left (3 A b^2+a^2 (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 a d}+\frac {2 \left (5 A b^2+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {10 A b (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
10/99*A*b*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d+2/11*A*(a+b *cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2)*sin(d*x+c)/d-2/693*(4*A*b^4-15*a^4*(9 *A+11*C)-a^2*b^2*(205*A+297*C))*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*cos(d*x+c ))^(1/2)/a^2/d+2/693*b*(3*A*b^2+a^2*(229*A+297*C))*sec(d*x+c)^(5/2)*sin(d* x+c)*(a+b*cos(d*x+c))^(1/2)/a/d+2/231*(5*A*b^2+3*a^2*(9*A+11*C))*sec(d*x+c )^(7/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d+2/693*(a-b)*b*(8*A*b^4+3*a^2*b ^2*(17*A+33*C)+a^4*(741*A+957*C))*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+b)^(1/2)*cos(d*x +c)^(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)+2/693*(a-b)*(6*a*A*b^3+8*A*b^4+15*a^4*(9*A+11*C)+3*a^ 2*b^2*(19*A+33*C)-6*a^3*b*(101*A+132*C))*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+b)^(1/2)* cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^( 1/2)/a^3/d/sec(d*x+c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(3885\) vs. \(2(627)=1254\).
Time = 24.23 (sec) , antiderivative size = 3885, normalized size of antiderivative = 6.20 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Result too large to show} \]
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*(741*a^4*A + 51*a^2*A*b ^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^2*C)*Sin[c + d*x])/(693*a^3) + (2*Sec[ c + d*x]^3*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[c + d*x] + 99*a^2*C*Sin[ c + d*x]))/693 + (2*Sec[c + d*x]^2*(229*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin [c + d*x] + 297*a^2*b*C*Sin[c + d*x]))/(693*a) + (2*Sec[c + d*x]*(135*a^4* A*Sin[c + d*x] + 205*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] + 165*a ^4*C*Sin[c + d*x] + 297*a^2*b^2*C*Sin[c + d*x]))/(693*a^2) + (46*a*A*b*Sec [c + d*x]^3*Tan[c + d*x])/99 + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/ d + (2*((-247*a^2*A*b)/(231*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (17*A*b^3)/(231*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^5)/ (693*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a^2*b*C)/(21*S qrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^3*C)/(7*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (15*a^3*A*Sqrt[Sec[c + d*x]])/(77*Sqrt[a + b *Cos[c + d*x]]) - (26*a*A*b^2*Sqrt[Sec[c + d*x]])/(231*Sqrt[a + b*Cos[c + d*x]]) - (7*A*b^4*Sqrt[Sec[c + d*x]])/(99*a*Sqrt[a + b*Cos[c + d*x]]) - (8 *A*b^6*Sqrt[Sec[c + d*x]])/(693*a^3*Sqrt[a + b*Cos[c + d*x]]) + (5*a^3*C*S qrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (2*a*b^2*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*C*Sqrt[Sec[c + d*x]])/(7*a*Sq rt[a + b*Cos[c + d*x]]) - (247*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]] )/(231*Sqrt[a + b*Cos[c + d*x]]) - (17*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[...
Time = 3.55 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.595, Rules used = {3042, 4709, 3042, 3527, 27, 3042, 3526, 27, 3042, 3526, 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.
| \(\displaystyle \int \sec ^{\frac {13}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{13/2} (a+b \cos (c+d x))^{5/2} \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^{5/2} \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {(a+b \cos (c+d x))^{3/2} \left (b (4 A+11 C) \cos ^2(c+d x)+a (9 A+11 C) \cos (c+d x)+5 A b\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {(a+b \cos (c+d x))^{3/2} \left (b (4 A+11 C) \cos ^2(c+d x)+a (9 A+11 C) \cos (c+d x)+5 A b\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (4 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (9 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b^2 (56 A+99 C) \cos ^2(c+d x)+2 a b (76 A+99 C) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+5 A b^2\right )\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b^2 (56 A+99 C) \cos ^2(c+d x)+2 a b (76 A+99 C) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+5 A b^2\right )\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b^2 (56 A+99 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (76 A+99 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+5 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {b \left (36 (9 A+11 C) a^2+b^2 (452 A+693 C)\right ) \cos ^2(c+d x)+a \left (45 (9 A+11 C) a^2+b^2 (1531 A+2079 C)\right ) \cos (c+d x)+5 b \left ((229 A+297 C) a^2+3 A b^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {b \left (36 (9 A+11 C) a^2+b^2 (452 A+693 C)\right ) \cos ^2(c+d x)+a \left (45 (9 A+11 C) a^2+b^2 (1531 A+2079 C)\right ) \cos (c+d x)+5 b \left ((229 A+297 C) a^2+3 A b^2\right )}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {b \left (36 (9 A+11 C) a^2+b^2 (452 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (45 (9 A+11 C) a^2+b^2 (1531 A+2079 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 b \left ((229 A+297 C) a^2+3 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \int -\frac {5 \left (-2 b^2 \left ((229 A+297 C) a^2+3 A b^2\right ) \cos ^2(c+d x)-a b \left (3 (337 A+429 C) a^2+b^2 (461 A+693 C)\right ) \cos (c+d x)+3 \left (-15 (9 A+11 C) a^4-b^2 (205 A+297 C) a^2+4 A b^4\right )\right )}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b^2 \left ((229 A+297 C) a^2+3 A b^2\right ) \cos ^2(c+d x)-a b \left (3 (337 A+429 C) a^2+b^2 (461 A+693 C)\right ) \cos (c+d x)+3 \left (-15 (9 A+11 C) a^4-b^2 (205 A+297 C) a^2+4 A b^4\right )}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b^2 \left ((229 A+297 C) a^2+3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a b \left (3 (337 A+429 C) a^2+b^2 (461 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-15 (9 A+11 C) a^4-b^2 (205 A+297 C) a^2+4 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {3 \left (b \left ((741 A+957 C) a^4+3 b^2 (17 A+33 C) a^2+8 A b^4\right )+a \left (15 (9 A+11 C) a^4+3 b^2 (221 A+297 C) a^2+2 A b^4\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {b \left ((741 A+957 C) a^4+3 b^2 (17 A+33 C) a^2+8 A b^4\right )+a \left (15 (9 A+11 C) a^4+3 b^2 (221 A+297 C) a^2+2 A b^4\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {b \left ((741 A+957 C) a^4+3 b^2 (17 A+33 C) a^2+8 A b^4\right )+a \left (15 (9 A+11 C) a^4+3 b^2 (221 A+297 C) a^2+2 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {b \left (a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+(a-b) \left (15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{a}}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {b \left (a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {b \left (a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \left (15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{a}\right )+\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \left (3 a^2 (9 A+11 C)+5 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 b \left (a^2 (229 A+297 C)+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \left (-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 b (a-b) \sqrt {a+b} \left (a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{a}\right )\right )+\frac {10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + b*Cos[c + d*x])^(5/2)*Sin [c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((10*A*b*(a + b*Cos[c + d*x])^(3/2 )*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*(5*A*b^2 + 3*a^2*(9*A + 11* C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2* b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/( a*d*Cos[c + d*x]^(5/2)) - (-(((2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^ 2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cot[c + d*x]*EllipticE[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^2*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11 *C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*Cot[c + d*x]*Elli pticF[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*(4*A*b^4 - 15*a^4*(9*A + 11*C) - a^2*b^ 2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Cos[c + d*x ]^(3/2)))/a)/7)/9)/11)
3.15.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[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_.) + (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/(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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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]
Timed out.
hanged
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
integral((C*b^2*cos(d*x + c)^4 + 2*C*a*b*cos(d*x + c)^3 + 2*A*a*b*cos(d*x + c) + A*a^2 + (C*a^2 + A*b^2)*cos(d*x + c)^2)*sqrt(b*cos(d*x + c) + a)*se c(d*x + c)^(13/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]