Integrand size = 41, antiderivative size = 510 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \] Output:
-2/3465*(110*B*a^4*b-3069*B*a^2*b^3-1617*B*b^5-40*C*a^5-15*a^3*b^2*(33*A+1 7*C)-15*a*b^4*(319*A+247*C))*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+ 1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/346 5*(a^2-b^2)*(110*B*a^3*b-1254*B*a*b^3-40*a^4*C-75*b^4*(11*A+9*C)-15*a^2*b^ 2*(33*A+19*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2* c,2^(1/2)*(b/(a+b))^(1/2))/b^3/d/(a+b*cos(d*x+c))^(1/2)-2/3465*(110*B*a^3* b-1254*B*a*b^3-40*a^4*C-75*b^4*(11*A+9*C)-15*a^2*b^2*(33*A+19*C))*(a+b*cos (d*x+c))^(1/2)*sin(d*x+c)/b^2/d-2/3465*(110*B*a^2*b-539*B*b^3-40*a^3*C-5*a *b^2*(99*A+67*C))*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+2/693*(99*A*b^2- 22*B*a*b+8*C*a^2+81*C*b^2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^2/d+2/99*(1 1*B*b-4*C*a)*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b^2/d+2/11*C*cos(d*x+c)*(a+ b*cos(d*x+c))^(7/2)*sin(d*x+c)/b/d
Time = 4.16 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+75 b^4 (11 A+9 C)+15 a^2 b^2 (297 A+221 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-110 a^4 b B+3069 a^2 b^3 B+1617 b^5 B+40 a^5 C+15 a^3 b^2 (33 A+17 C)+15 a b^4 (319 A+247 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (880 a^3 b B+32868 a b^3 B-320 a^4 C+60 a^2 b^2 (396 A+311 C)+30 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 b B+1463 b^3 B+30 a^3 C+5 a b^2 (594 A+619 C)\right ) \sin (2 (c+d x))+5 b \left (\left (396 A b^2+836 a b B+452 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((22 b B+46 a C) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )}{27720 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C* Cos[c + d*x]^2),x]
Output:
(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(1705*a^3*b*B + 2871*a*b^3*B + 10*a^4*C + 75*b^4*(11*A + 9*C) + 15*a^2*b^2*(297*A + 221*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 15*a^3*b^2*(33*A + 17*C) + 15*a*b^4*(319*A + 247*C))*((a + b)*E llipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((880*a^3*b*B + 32868*a*b^3*B - 320*a^4*C + 60*a^2*b^2*(396*A + 311*C) + 30*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4 *(1650*a^2*b*B + 1463*b^3*B + 30*a^3*C + 5*a*b^2*(594*A + 619*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 836*a*b*B + 452*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*((22*b*B + 46*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)]) ))))/(27720*b^3*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.83 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.04, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.585, Rules used = {3042, 3528, 27, 3042, 3502, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} \left ((11 b B-4 a C) \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+2 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{5/2} \left ((11 b B-4 a C) \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+2 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left ((11 b B-4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (b (77 b B-10 a C)+\left (8 C a^2-22 b B a+99 A b^2+81 b^2 C\right ) \cos (c+d x)\right )dx}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (a+b \cos (c+d x))^{5/2} \left (b (77 b B-10 a C)+\left (8 C a^2-22 b B a+99 A b^2+81 b^2 C\right ) \cos (c+d x)\right )dx}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (b (77 b B-10 a C)+\left (8 C a^2-22 b B a+99 A b^2+81 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b \left (-10 C a^2+143 b B a+165 A b^2+135 b^2 C\right )-\left (-40 C a^3+110 b B a^2-5 b^2 (99 A+67 C) a-539 b^3 B\right ) \cos (c+d x)\right )dx+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (3 b \left (-10 C a^2+143 b B a+165 A b^2+135 b^2 C\right )-\left (-40 C a^3+110 b B a^2-5 b^2 (99 A+67 C) a-539 b^3 B\right ) \cos (c+d x)\right )dx+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (-10 C a^2+143 b B a+165 A b^2+135 b^2 C\right )+\left (40 C a^3-110 b B a^2+5 b^2 (99 A+67 C) a+539 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-10 C a^3+605 b B a^2+10 b^2 (132 A+101 C) a+539 b^3 B\right )-\left (-40 C a^4+110 b B a^3-15 b^2 (33 A+19 C) a^2-1254 b^3 B a-75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (-10 C a^3+605 b B a^2+10 b^2 (132 A+101 C) a+539 b^3 B\right )-\left (-40 C a^4+110 b B a^3-15 b^2 (33 A+19 C) a^2-1254 b^3 B a-75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-10 C a^3+605 b B a^2+10 b^2 (132 A+101 C) a+539 b^3 B\right )+\left (40 C a^4-110 b B a^3+15 b^2 (33 A+19 C) a^2+1254 b^3 B a+75 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {b \left (10 C a^4+1705 b B a^3+15 b^2 (297 A+221 C) a^2+2871 b^3 B a+75 b^4 (11 A+9 C)\right )-\left (-40 C a^5+110 b B a^4-15 b^2 (33 A+17 C) a^3-3069 b^3 B a^2-15 b^4 (319 A+247 C) a-1617 b^5 B\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (10 C a^4+1705 b B a^3+15 b^2 (297 A+221 C) a^2+2871 b^3 B a+75 b^4 (11 A+9 C)\right )-\left (-40 C a^5+110 b B a^4-15 b^2 (33 A+17 C) a^3-3069 b^3 B a^2-15 b^4 (319 A+247 C) a-1617 b^5 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (10 C a^4+1705 b B a^3+15 b^2 (297 A+221 C) a^2+2871 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (40 C a^5-110 b B a^4+15 b^2 (33 A+17 C) a^3+3069 b^3 B a^2+15 b^4 (319 A+247 C) a+1617 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}\right )}{9 b}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
Input:
Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d) + ((2* (11*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d) + ((2*(9 9*A*b^2 - 22*a*b*B + 8*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((-2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 5*a*b^2*(99*A + 67*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (3*(((-2*(110*a^4 *b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 15*a^3*b^2*(33*A + 17*C) - 15*a*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2) *(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2 *(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/3 - (2*(110*a^3*b*B - 1254 *a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt [a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/7)/(9*b))/(11*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(2602\) vs. \(2(487)=974\).
Time = 86.00 (sec) , antiderivative size = 2603, normalized size of antiderivative = 5.10
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2603\) |
| parts | \(\text {Expression too large to display}\) | \(2964\) |
Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
-2/3465*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(40*C* (sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b)) ^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6+675*C*b^6*(sin (1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/ 2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-40*C*(sin(1/2*d*x+1/2* c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF( cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6-1617*B*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^6+825*A*b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2) *(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x +1/2*c),(-2*b/(a-b))^(1/2))+1617*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b )*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2 *b/(a-b))^(1/2))*a*b^5-245*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin( 1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a- b))^(1/2))*a^4*b^2-495*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2* d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^ (1/2))*a^3*b^3-390*a^2*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2* d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^ (1/2))*b^4+495*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2* c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.59 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="fricas")
Output:
-2/10395*(sqrt(1/2)*(-80*I*C*a^6 + 220*I*B*a^5*b - 30*I*(33*A + 16*C)*a^4* b^2 - 1023*I*B*a^3*b^3 + 15*I*(253*A + 169*C)*a^2*b^4 + 5379*I*B*a*b^5 + 2 25*I*(11*A + 9*C)*b^6)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2 , -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(80*I*C*a^6 - 220*I*B*a^5*b + 30*I*(33*A + 16*C)*a^4 *b^2 + 1023*I*B*a^3*b^3 - 15*I*(253*A + 169*C)*a^2*b^4 - 5379*I*B*a*b^5 - 225*I*(11*A + 9*C)*b^6)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^ 2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(-40*I*C*a^5*b + 110*I*B*a^4*b^2 - 15*I*(33*A + 1 7*C)*a^3*b^3 - 3069*I*B*a^2*b^4 - 15*I*(319*A + 247*C)*a*b^5 - 1617*I*B*b^ 6)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2 )/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2 )/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sqrt(1/2) *(40*I*C*a^5*b - 110*I*B*a^4*b^2 + 15*I*(33*A + 17*C)*a^3*b^3 + 3069*I*B*a ^2*b^4 + 15*I*(319*A + 247*C)*a*b^5 + 1617*I*B*b^6)*sqrt(b)*weierstrassZet a(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInvers e(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(315*C*b^6*cos(d*x + c)^4 - 20*C*a ^4*b^2 + 55*B*a^3*b^3 + 5*(297*A + 205*C)*a^2*b^4 + 1793*B*a*b^5 + 75*(11* A + 9*C)*b^6 + 35*(23*C*a*b^5 + 11*B*b^6)*cos(d*x + c)^3 + 5*(113*C*a^2...
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)* *2),x)
Output:
Timed out
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^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)*cos(d*x + c), x)
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^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)*cos(d*x + c), x)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int(cos(c + d*x)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}d x \right ) b^{2} c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) a b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} c +3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b \] Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x),x)*a**3 + int(sqrt(cos(c + d*x)* b + a)*cos(c + d*x)**5,x)*b**2*c + 2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4,x)*a*b*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4,x)*b**3 + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*a**2*c + 3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*a*b**2 + 3*int(sqrt(cos(c + d*x)*b + a)*c os(c + d*x)**2,x)*a**2*b