Integrand size = 35, antiderivative size = 443 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {4 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1155 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{1155 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{1155 b^3 d}-\frac {4 a \left (33 A b^2+8 a^2 C+34 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{1155 b^3 d}+\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{231 b^3 d}-\frac {4 a C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{33 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d} \] Output:
-4/1155*a*(8*a^4*C+3*a^2*b^2*(11*A+6*C)-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)+2/1155*(a^2-b^2)*(16*a^4*C+6*a^2*b^2*(11*A+8*C)-2 5*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)-2/1155*(16*a^4* C+6*a^2*b^2*(11*A+8*C)-25*b^4*(11*A+9*C))*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c )/b^3/d-4/1155*a*(33*A*b^2+8*C*a^2+34*C*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d* x+c)/b^3/d+2/231*(8*a^2*C+3*b^2*(11*A+9*C))*(a+b*cos(d*x+c))^(5/2)*sin(d*x +c)/b^3/d-4/33*a*C*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 = 3.33 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.75 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (-4 a^4 b C+25 b^5 (11 A+9 C)+3 a^2 b^3 (187 A+141 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (2 \left (64 a^4 C+6 a^2 b^2 (44 A+27 C)+5 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (16 a \left (132 A b^2-3 a^2 C+136 b^2 C\right ) \sin (2 (c+d x))+5 b \left (\left (132 A b^2+4 a^2 C+171 b^2 C\right ) \sin (3 (c+d x))+7 b C (8 a \sin (4 (c+d x))+3 b \sin (5 (c+d x)))\right )\right )\right )}{9240 b^4 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2) ,x]
Output:
(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(-4*a^4*b*C + 25*b^5*(11*A + 9*C ) + 3*a^2*b^3*(187*A + 141*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a *(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*((a + b)*Ellipti cE[(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*(64*a^4*C + 6*a^2*b^2*(44*A + 27*C) + 5*b^4*( 506*A + 435*C))*Sin[c + d*x] + b*(16*a*(132*A*b^2 - 3*a^2*C + 136*b^2*C)*S in[2*(c + d*x)] + 5*b*((132*A*b^2 + 4*a^2*C + 171*b^2*C)*Sin[3*(c + d*x)] + 7*b*C*(8*a*Sin[4*(c + d*x)] + 3*b*Sin[5*(c + d*x)])))))/(9240*b^4*d*Sqrt [a + b*Cos[c + d*x]])
Time = 2.64 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {3042, 3529, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (-6 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+4 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (-6 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+4 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-6 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int -\frac {3}{2} (a+b \cos (c+d x))^{3/2} \left (4 C a^2+2 b C \cos (c+d x) a-\left (8 C a^2+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )dx}{9 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int (a+b \cos (c+d x))^{3/2} \left (4 C a^2+2 b C \cos (c+d x) a-\left (8 C a^2+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )dx}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (4 C a^2+2 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-8 C a^2-3 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {-\frac {\frac {2 \int -\frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b \left (4 C a^2+55 A b^2+45 b^2 C\right )-2 a \left (8 C a^2+33 A b^2+34 b^2 C\right ) \cos (c+d x)\right )dx}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int (a+b \cos (c+d x))^{3/2} \left (3 b \left (4 C a^2+55 A b^2+45 b^2 C\right )-2 a \left (8 C a^2+33 A b^2+34 b^2 C\right ) \cos (c+d x)\right )dx}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (4 C a^2+55 A b^2+45 b^2 C\right )-2 a \left (8 C a^2+33 A b^2+34 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (a b \left (4 C a^2+209 A b^2+157 b^2 C\right )-\left (16 C a^4+6 b^2 (11 A+8 C) a^2-25 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (a b \left (4 C a^2+209 A b^2+157 b^2 C\right )-\left (16 C a^4+6 b^2 (11 A+8 C) a^2-25 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (a b \left (4 C a^2+209 A b^2+157 b^2 C\right )+\left (-16 C a^4-6 b^2 (11 A+8 C) a^2+25 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {2}{3} \int -\frac {b \left (4 C a^4-3 b^2 (187 A+141 C) a^2-25 b^4 (11 A+9 C)\right )+2 a \left (8 C a^4+3 b^2 (11 A+6 C) a^2-b^4 (451 A+348 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (4 C a^4-3 b^2 (187 A+141 C) a^2-25 b^4 (11 A+9 C)\right )+2 a \left (8 C a^4+3 b^2 (11 A+6 C) a^2-b^4 (451 A+348 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (4 C a^4-3 b^2 (187 A+141 C) a^2-25 b^4 (11 A+9 C)\right )+2 a \left (8 C a^4+3 b^2 (11 A+6 C) a^2-b^4 (451 A+348 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {4 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {-\frac {2 \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {4 a \left (8 a^2 C+33 A b^2+34 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}}{3 b}-\frac {4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{3 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}\) |
Input:
Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + 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) + (( -4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*b*d) - ((- 2*(8*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/ (7*b*d) - ((-4*a*(33*A*b^2 + 8*a^2*C + 34*b^2*C)*(a + b*Cos[c + d*x])^(3/2 )*Sin[c + d*x])/(5*d) + (3*(((-4*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - 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)*(16*a^4* C + 6*a^2*b^2*(11*A + 8*C) - 25*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*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*S qrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/(7*b))/(3*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])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, 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. \(1790\) vs. \(2(420)=840\).
Time = 17.19 (sec) , antiderivative size = 1791, normalized size of antiderivative = 4.04
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1791\) |
| parts | \(\text {Expression too large to display}\) | \(1969\) |
Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x,method=_RETUR NVERBOSE)
Output:
-2/1155*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(6720* C*b^6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-7840*C*a*b^5-16800*C*b^6) *sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(2640*A*b^6+2320*C*a^2*b^4+15680 *C*a*b^5+18960*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3432*A*a*b ^5-3960*A*b^6+8*C*a^3*b^3-3480*C*a^2*b^4-14456*C*a*b^5-11640*C*b^6)*sin(1/ 2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1188*A*a^2*b^4+3432*A*a*b^5+3080*A*b^6+ 8*C*a^4*b^2-8*C*a^3*b^3+2624*C*a^2*b^4+6616*C*a*b^5+4620*C*b^6)*sin(1/2*d* x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-66*A*a^3*b^3-594*A*a^2*b^4-1408*A*a*b^5-88 0*A*b^6-16*C*a^5*b-4*C*a^4*b^2-36*C*a^3*b^3-732*C*a^2*b^4-1614*C*a*b^5-930 *C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+66*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)*EllipticF(cos( 1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2-341*A*a^2*(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+275*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))-66*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^4*b^2+66*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*si n(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+902*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin...
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.51 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algori thm="fricas")
Output:
-2/3465*(sqrt(1/2)*(32*I*C*a^6 + 12*I*(11*A + 5*C)*a^4*b^2 - I*(121*A + 12 3*C)*a^2*b^4 + 75*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)*(-32*I*C*a^6 - 12*I*(11*A + 5*C)*a^4* b^2 + I*(121*A + 123*C)*a^2*b^4 - 75*I*(11*A + 9*C)*b^6)*sqrt(b)*weierstra ssPInverse(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) + 6*sqrt(1/2)*(8*I*C*a^5*b + 3 *I*(11*A + 6*C)*a^3*b^3 - I*(451*A + 348*C)*a*b^5)*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)) + 6*sqrt(1/2)*(-8*I*C*a^5*b - 3*I*(11*A + 6*C)*a^3*b^3 + I*(451*A + 348*C)*a*b^5)*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*(105*C*b^6*cos(d*x + c)^4 + 140*C*a*b^5*cos (d*x + c)^3 + 8*C*a^4*b^2 + (33*A + 19*C)*a^2*b^4 + 25*(11*A + 9*C)*b^6 + 5*(C*a^2*b^4 + 3*(11*A + 9*C)*b^6)*cos(d*x + c)^2 - 2*(3*C*a^3*b^3 - (132* A + 101*C)*a*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^ 5*d)
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**2),x)
Output:
Timed out
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(3/2)*cos(d*x + c)^2 , x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(3/2)*cos(d*x + c)^2 , x)
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2),x)
Output:
int(cos(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+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 )**3,x)*a*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a**2