Integrand size = 45, antiderivative size = 705 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (40 A b^4+10 a b^3 (3 A-11 B)+15 a^2 b^2 (19 A-121 B+33 C)+3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \] Output:
2/3465*(a-b)*(a+b)^(1/2)*(40*A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B+1 5*a^2*b^3*(17*A+33*C)+15*a^4*b*(247*A+319*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^ 4/d/sec(d*x+c)^(1/2)+2/3465*(a-b)*(a+b)^(1/2)*(40*A*b^4+10*a*b^3*(3*A-11*B )+15*a^2*b^2*(19*A-121*B+33*C)+3*a^4*(225*A-539*B+275*C)-6*a^3*b*(505*A-20 9*B+660*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^3/d/sec(d*x+c)^(1/2)-2/3465*(20*A* b^4-1793*B*a^3*b-55*B*a*b^3-75*a^4*(9*A+11*C)-5*a^2*b^2*(205*A+297*C))*(a+ b*cos(d*x+c))^(1/2)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^2/d+2/3465*(15*A*b^3+539 *B*a^3+825*B*a*b^2+5*a^2*b*(229*A+297*C))*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c )^(5/2)*sin(d*x+c)/a/d+2/231*(5*A*b^2+44*B*a*b+3*a^2*(9*A+11*C))*(a+b*cos( d*x+c))^(1/2)*sec(d*x+c)^(7/2)*sin(d*x+c)/d+2/99*(5*A*b+11*B*a)*(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)*s ec(d*x+c)^(11/2)*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(5243\) vs. \(2(705)=1410\).
Time = 28.69 (sec) , antiderivative size = 5243, normalized size of antiderivative = 7.44 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^(13/2),x]
Output:
Result too large to show
Time = 4.27 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.01, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {3042, 4709, 3042, 3526, 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.
| \(\displaystyle \int \sec ^{\frac {13}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{13/2} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^{5/2} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {(a+b \cos (c+d x))^{3/2} \left (b (4 A+11 C) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+5 A b+11 a B\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {(a+b \cos (c+d x))^{3/2} \left (b (4 A+11 C) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+5 A b+11 a B\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (4 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 A b+11 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b (56 A b+99 C b+44 a B) \cos ^2(c+d x)+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \cos (c+d x)} \left (b (56 A b+99 C b+44 a B) \cos ^2(c+d x)+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (56 A b+99 C b+44 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \cos ^2(c+d x)+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \cos (c+d x)}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \cos ^2(c+d x)+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \int -\frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \cos ^2(c+d x)-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \cos (c+d x)+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \cos ^2(c+d x)-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \cos (c+d x)+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {3 \left (1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \cos (c+d x) a+40 A b^5\right )}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \cos (c+d x) a+40 A b^5}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+40 A b^5}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)+15 a^2 b^2 (19 A-121 B+33 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {(a-b) \left (3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)+15 a^2 b^2 (19 A-121 B+33 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)+15 a^2 b^2 (19 A-121 B+33 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)+15 a^2 b^2 (19 A-121 B+33 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}}{a}}{5 a}\right )\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + 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 + b*Cos[c + d*x])^(5/2)*Sin [c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2*(5*A*b + 11*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 + 44*a *b*B + 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*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229* A + 297*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]*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*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]*(40*A*b^4 + 10*a*b^3*(3*A - 11*B) + 15*a^2*b^2*(19*A - 121*B + 33*C) + 3*a^4*(225*A - 539*B + 275*C) - 6*a^3*b*(505*A - 209*B + 660*C))*Cot[c + d*x]*EllipticF [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*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a ^4*(9*A + 11*C) - 5*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)))/(5*a))/7)/9)/11)
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]
Timed out.
hanged
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/ 2),x)
Output:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/ 2),x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(13/2),x, algorithm="fricas")
Output:
integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c ))*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(13/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**(13/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(13/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c)^(13/2), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(13/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c)^(13/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int((1/cos(c + d*x))^(13/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int((1/cos(c + d*x))^(13/2)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{6}d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{6}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{6}d x \right ) a^{2} c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{6}d x \right ) a \,b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{6}d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/ 2),x)
Output:
3*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x )**6,x)*a**2*b + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d *x)**4*sec(c + d*x)**6,x)*b**2*c + 2*int(sqrt(sec(c + d*x))*sqrt(cos(c + d *x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**6,x)*a*b*c + int(sqrt(sec(c + d*x ))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**6,x)*b**3 + int( sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)** 6,x)*a**2*c + 3*int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x)*b + a)*cos(c + d* x)**2*sec(c + d*x)**6,x)*a*b**2 + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x) *b + a)*sec(c + d*x)**6,x)*a**3