Integrand size = 43, antiderivative size = 518 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d} \] Output:
2/3465*(88*B*a^4*b+363*B*a^2*b^3+1617*B*b^5-48*C*a^5-18*a^3*b^2*(11*A+6*C) +6*a*b^4*(451*A+348*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^4/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/3465*(a^ 2-b^2)*(88*B*a^3*b+429*B*a*b^3-48*a^4*C-18*a^2*b^2*(11*A+8*C)+75*b^4*(11*A +9*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^4/d/(a+b*cos(d*x+c))^(1/2)+2/3465*(88*B*a^3*b+429*B*a *b^3-48*a^4*C-18*a^2*b^2*(11*A+8*C)+75*b^4*(11*A+9*C))*(a+b*cos(d*x+c))^(1 /2)*sin(d*x+c)/b^3/d+2/3465*(88*B*a^2*b+539*B*b^3-48*a^3*C-6*a*b^2*(33*A+3 4*C))*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^3/d+2/693*(99*A*b^2-44*B*a*b+24* C*a^2+81*C*b^2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^3/d+2/99*(11*B*b-6*C*a )*cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^2/d+2/11*C*cos(d*x+c)^2*( a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d
Time = 4.27 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.79 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-88 a^4 b B-363 a^2 b^3 B-1617 b^5 B+48 a^5 C+18 a^3 b^2 (11 A+6 C)-6 a b^4 (451 A+348 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (2 \left (-352 a^3 b B+8844 a b^3 B+192 a^4 C+18 a^2 b^2 (44 A+27 C)+15 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (4 \left (66 a^2 b B+1463 b^3 B-36 a^3 C+48 a b^2 (33 A+34 C)\right ) \sin (2 (c+d x))+5 b \left (\left (396 A b^2+440 a b B+12 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((22 b B+24 a C) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )}{27720 b^4 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(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*(22*a^3*b*B + 2046*a*b^3*B - 1 2*a^4*C + 75*b^4*(11*A + 9*C) + 9*a^2*b^2*(187*A + 141*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^ 5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*((a + b)*Elliptic E[(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])*(2*(-352*a^3*b*B + 8844*a*b^3*B + 192*a^4*C + 18* a^2*b^2*(44*A + 27*C) + 15*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*(66*a^ 2*b*B + 1463*b^3*B - 36*a^3*C + 48*a*b^2*(33*A + 34*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 440*a*b*B + 12*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7* b*((22*b*B + 24*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(27720 *b^4*d*Sqrt[a + b*Cos[c + d*x]])
Time = 3.02 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left ((11 b B-6 a C) \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+4 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left ((11 b B-6 a C) \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+4 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((11 b B-6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (\left (24 C a^2-44 b B a+99 A b^2+81 b^2 C\right ) \cos ^2(c+d x)+b (77 b B-6 a C) \cos (c+d x)+2 a (11 b B-6 a C)\right )dx}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (a+b \cos (c+d x))^{3/2} \left (\left (24 C a^2-44 b B a+99 A b^2+81 b^2 C\right ) \cos ^2(c+d x)+b (77 b B-6 a C) \cos (c+d x)+2 a (11 b B-6 a C)\right )dx}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\left (24 C a^2-44 b B a+99 A b^2+81 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (77 b B-6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a (11 b B-6 a C)\right )dx}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b \left (12 C a^2-22 b B a+165 A b^2+135 b^2 C\right )+\left (-48 C a^3+88 b B a^2-6 b^2 (33 A+34 C) a+539 b^3 B\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int (a+b \cos (c+d x))^{3/2} \left (3 b \left (12 C a^2-22 b B a+165 A b^2+135 b^2 C\right )+\left (-48 C a^3+88 b B a^2-6 b^2 (33 A+34 C) a+539 b^3 B\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (12 C a^2-22 b B a+165 A b^2+135 b^2 C\right )+\left (-48 C a^3+88 b B a^2-6 b^2 (33 A+34 C) a+539 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {\frac {2}{5} \int -\frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-12 C a^3+22 b B a^2-3 b^2 (209 A+157 C) a-539 b^3 B\right )-\left (-48 C a^4+88 b B a^3-18 b^2 (11 A+8 C) a^2+429 b^3 B a+75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (-12 C a^3+22 b B a^2-3 b^2 (209 A+157 C) a-539 b^3 B\right )-\left (-48 C a^4+88 b B a^3-18 b^2 (11 A+8 C) a^2+429 b^3 B a+75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-12 C a^3+22 b B a^2-3 b^2 (209 A+157 C) a-539 b^3 B\right )+\left (48 C a^4-88 b B a^3+18 b^2 (11 A+8 C) a^2-429 b^3 B a-75 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {2}{3} \int -\frac {b \left (-12 C a^4+22 b B a^3+9 b^2 (187 A+141 C) a^2+2046 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (-48 C a^5+88 b B a^4-18 b^2 (11 A+6 C) a^3+363 b^3 B a^2+6 b^4 (451 A+348 C) a+1617 b^5 B\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (-12 C a^4+22 b B a^3+9 b^2 (187 A+141 C) a^2+2046 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (-48 C a^5+88 b B a^4-18 b^2 (11 A+6 C) a^3+363 b^3 B a^2+6 b^4 (451 A+348 C) a+1617 b^5 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (-12 C a^4+22 b B a^3+9 b^2 (187 A+141 C) a^2+2046 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (-48 C a^5+88 b B a^4-18 b^2 (11 A+6 C) a^3+363 b^3 B a^2+6 b^4 (451 A+348 C) a+1617 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{7 b d}+\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}}{9 b}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
Input:
Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[ c + d*x]^2),x]
Output:
(2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d) + (( 2*(11*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/( 9*b*d) + ((2*(99*A*b^2 - 44*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*Cos[c + d* x])^(5/2)*Sin[c + d*x])/(7*b*d) + ((2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 6*a*b^2*(33*A + 34*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) - ( 3*(((-2*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*( 11*A + 6*C) + 6*a*b^4*(451*A + 348*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)*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C ) + 75*b^4*(11*A + 9*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*(88*a^3*b* B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C) )*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/(7*b))/(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. \(2(495)=990\).
Time = 62.24 (sec) , antiderivative size = 2603, normalized size of antiderivative = 5.03
| 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)^2*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_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)*(-48*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*(si n(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))+48*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))+(-12320*B*b^6-23520*C*a*b^5-50400*C*b^6)*sin( 1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^6+14960*B*a*b^5+24640*B*b^6 +6960*C*a^2*b^4+47040*C*a*b^5+56880*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d* x+1/2*c)+(-10296*A*a*b^5-11880*A*b^6-4664*B*a^2*b^4-22440*B*a*b^5-22792*B* b^6+24*C*a^3*b^3-10440*C*a^2*b^4-43368*C*a*b^5-34920*C*b^6)*sin(1/2*d*x+1/ 2*c)^6*cos(1/2*d*x+1/2*c)+(3564*A*a^2*b^4+10296*A*a*b^5+9240*A*b^6-44*B*a^ 3*b^3+4664*B*a^2*b^4+17248*B*a*b^5+10472*B*b^6+24*C*a^4*b^2-24*C*a^3*b^3+7 872*C*a^2*b^4+19848*C*a*b^5+13860*C*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+ 1/2*c)+(-198*A*a^3*b^3-1782*A*a^2*b^4-4224*A*a*b^5-2640*A*b^6+88*B*a^4*b^2 +22*B*a^3*b^3-3102*B*a^2*b^4-4884*B*a*b^5-1848*B*b^6-48*C*a^5*b-12*C*a^...
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.57 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
Output:
-2/10395*(sqrt(1/2)*(96*I*C*a^6 - 176*I*B*a^5*b + 36*I*(11*A + 5*C)*a^4*b^ 2 - 660*I*B*a^3*b^3 - 3*I*(121*A + 123*C)*a^2*b^4 + 2904*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) + sqrt(1/2)*(-96*I*C*a^6 + 176*I*B*a^5*b - 36*I*(11*A + 5*C)*a^4*b^2 + 660*I*B*a^3*b^3 + 3*I*(121*A + 123*C)*a^2*b^4 - 2904*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)*(48*I*C*a^5*b - 88*I*B*a^4*b^2 + 18*I*(11*A + 6*C)*a^3* b^3 - 363*I*B*a^2*b^4 - 6*I*(451*A + 348*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, weie rstrassPInverse(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)*(-48*I*C*a ^5*b + 88*I*B*a^4*b^2 - 18*I*(11*A + 6*C)*a^3*b^3 + 363*I*B*a^2*b^4 + 6*I* (451*A + 348*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*si n(d*x + c) + 2*a)/b)) - 3*(315*C*b^6*cos(d*x + c)^4 + 24*C*a^4*b^2 - 44*B* a^3*b^3 + 3*(33*A + 19*C)*a^2*b^4 + 968*B*a*b^5 + 75*(11*A + 9*C)*b^6 + 35 *(12*C*a*b^5 + 11*B*b^6)*cos(d*x + c)^3 + 5*(3*C*a^2*b^4 + 110*B*a*b^5 ...
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+ c)**2),x)
Output:
Timed out
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(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)^(3/ 2)*cos(d*x + c)^2, x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(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)^(3/ 2)*cos(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos( c + d*x)^2),x)
Output:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos( c + d*x)^2), x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) b^{2}+2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(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)**5,x)*b*c + int(sqrt(cos(c + d*x )*b + a)*cos(c + d*x)**4,x)*a*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x )**4,x)*b**2 + 2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*a*b + int (sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a**2