Integrand size = 43, antiderivative size = 416 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (24 a^3 b B+57 a b^3 B-16 a^4 C-6 a^2 b^2 (7 A+4 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (24 a^2 b B+75 b^3 B-16 a^3 C-6 a b^2 (7 A+6 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (24 a^2 b B+75 b^3 B-16 a^3 C-6 a b^2 (7 A+6 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}+\frac {2 \left (63 A b^2-36 a b B+24 a^2 C+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac {2 (3 b B-2 a C) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d} \] Output:
2/315*(24*B*a^3*b+57*B*a*b^3-16*a^4*C-6*a^2*b^2*(7*A+4*C)+21*b^4*(9*A+7*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)-2/315*(a^2-b^2)*(24*B*a^2*b+75*B *b^3-16*a^3*C-6*a*b^2*(7*A+6*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJac obiAM(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)+ 2/315*(24*B*a^2*b+75*B*b^3-16*a^3*C-6*a*b^2*(7*A+6*C))*(a+b*cos(d*x+c))^(1 /2)*sin(d*x+c)/b^3/d+2/315*(63*A*b^2-36*B*a*b+24*C*a^2+49*C*b^2)*(a+b*cos( d*x+c))^(3/2)*sin(d*x+c)/b^3/d+2/21*(3*B*b-2*C*a)*cos(d*x+c)*(a+b*cos(d*x+ c))^(3/2)*sin(d*x+c)/b^2/d+2/9*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*sin(d *x+c)/b/d
Time = 2.66 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (6 a^2 b B+75 b^3 B-4 a^3 C+3 a b^2 (49 A+37 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-24 a^3 b B-57 a b^3 B+16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (2 \left (-48 a^2 b B+345 b^3 B+32 a^3 C+3 a b^2 (28 A+19 C)\right ) \sin (c+d x)+b \left (\left (252 A b^2+36 a b B-24 a^2 C+266 b^2 C\right ) \sin (2 (c+d x))+5 b (2 (9 b B+a C) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right )\right )}{1260 b^4 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C* Cos[c + d*x]^2),x]
Output:
(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(6*a^2*b*B + 75*b^3*B - 4*a^3*C + 3*a*b^2*(49*A + 37*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-24*a^3 *b*B - 57*a*b^3*B + 16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C)) *((a + b)*EllipticE[(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*(-48*a^2*b*B + 345*b^3*B + 3 2*a^3*C + 3*a*b^2*(28*A + 19*C))*Sin[c + d*x] + b*((252*A*b^2 + 36*a*b*B - 24*a^2*C + 266*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + a*C)*Sin[3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^4*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.35 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.05, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (3 (3 b B-2 a C) \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+4 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (3 (3 b B-2 a C) \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+4 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 (3 b B-2 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (\left (24 C a^2-36 b B a+63 A b^2+49 b^2 C\right ) \cos ^2(c+d x)+b (45 b B-2 a C) \cos (c+d x)+6 a (3 b B-2 a C)\right )dx}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \sqrt {a+b \cos (c+d x)} \left (\left (24 C a^2-36 b B a+63 A b^2+49 b^2 C\right ) \cos ^2(c+d x)+b (45 b B-2 a C) \cos (c+d x)+6 a (3 b B-2 a C)\right )dx}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (24 C a^2-36 b B a+63 A b^2+49 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (45 b B-2 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 a (3 b B-2 a C)\right )dx}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (4 C a^2-6 b B a+63 A b^2+49 b^2 C\right )+\left (-16 C a^3+24 b B a^2-6 b^2 (7 A+6 C) a+75 b^3 B\right ) \cos (c+d x)\right )dx}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \sqrt {a+b \cos (c+d x)} \left (b \left (4 C a^2-6 b B a+63 A b^2+49 b^2 C\right )+\left (-16 C a^3+24 b B a^2-6 b^2 (7 A+6 C) a+75 b^3 B\right ) \cos (c+d x)\right )dx}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (4 C a^2-6 b B a+63 A b^2+49 b^2 C\right )+\left (-16 C a^3+24 b B a^2-6 b^2 (7 A+6 C) a+75 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2}{3} \int \frac {b \left (-4 C a^3+6 b B a^2+3 b^2 (49 A+37 C) a+75 b^3 B\right )+\left (-16 C a^4+24 b B a^3-6 b^2 (7 A+4 C) a^2+57 b^3 B a+21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \int \frac {b \left (-4 C a^3+6 b B a^2+3 b^2 (49 A+37 C) a+75 b^3 B\right )+\left (-16 C a^4+24 b B a^3-6 b^2 (7 A+4 C) a^2+57 b^3 B a+21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \int \frac {b \left (-4 C a^3+6 b B a^2+3 b^2 (49 A+37 C) a+75 b^3 B\right )+\left (-16 C a^4+24 b B a^3-6 b^2 (7 A+4 C) a^2+57 b^3 B a+21 b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {2 \left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {2 \left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{3} \left (\frac {2 \left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \sin (c+d x) \left (24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 b d}+\frac {3 \left (\frac {2 \sin (c+d x) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}\right )\right )}{5 b}}{7 b}+\frac {6 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
Input:
Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(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])^(3/2)*Sin[c + d*x])/(9*b*d) + ((6 *(3*b*B - 2*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7* b*d) + ((2*(63*A*b^2 - 36*a*b*B + 24*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x] )^(3/2)*Sin[c + d*x])/(5*b*d) + (3*(((2*(24*a^3*b*B + 57*a*b^3*B - 16*a^4* C - 6*a^2*b^2*(7*A + 4*C) + 21*b^4*(9*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*E llipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(24*a^2*b*B + 75*b^3*B - 16*a^3*C - 6*a*b^2*(7*A + 6 *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*(24*a^2*b*B + 75*b^3*B - 16*a^ 3*C - 6*a*b^2*(7*A + 6*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/ (5*b))/(7*b))/(9*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. \(2142\) vs. \(2(397)=794\).
Time = 37.08 (sec) , antiderivative size = 2143, normalized size of antiderivative = 5.15
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2143\) |
| parts | \(\text {Expression too large to display}\) | \(2489\) |
Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
-2/315*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(16*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^5+75*B*b^5*(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))-189*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(c os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^5-147*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))*b^5-16*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^5-1120*C*b^5*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^ 10-57*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^4+42*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^3*b^2-42*a*A*(s in(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^4-24*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)*Elli pticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^2+24*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)*Ellipt...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.68 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
Output:
-2/945*(sqrt(1/2)*(32*I*C*a^5 - 48*I*B*a^4*b + 12*I*(7*A + 3*C)*a^3*b^2 - 96*I*B*a^2*b^3 + 3*I*(21*A + 13*C)*a*b^4 + 225*I*B*b^5)*sqrt(b)*weierstras sPInverse(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*c os(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(-32*I*C*a^5 + 48*I *B*a^4*b - 12*I*(7*A + 3*C)*a^3*b^2 + 96*I*B*a^2*b^3 - 3*I*(21*A + 13*C)*a *b^4 - 225*I*B*b^5)*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)*(16*I*C*a^4*b - 24*I*B*a^3*b^2 + 6*I*(7*A + 4*C)*a^2* b^3 - 57*I*B*a*b^4 - 21*I*(9*A + 7*C)*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*sqrt(1/2)*(-16*I*C*a^4*b + 24*I*B*a^3*b^2 - 6*I*(7*A + 4*C)*a^2*b^3 + 57*I*B*a*b^4 + 21*I*(9*A + 7*C)*b^5)*sqrt(b)*we ierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weiers trassPInverse(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*(35*C*b^5*cos(d*x + c) ^3 + 8*C*a^3*b^2 - 12*B*a^2*b^3 + (21*A + 13*C)*a*b^4 + 75*B*b^5 + 5*(C*a* b^4 + 9*B*b^5)*cos(d*x + c)^2 - (6*C*a^2*b^3 - 9*B*a*b^4 - 7*(9*A + 7*C)*b ^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^5*d)
Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+ c)**2),x)
Output:
Timed out
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(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)*sqrt(b*cos(d*x + c) + a) *cos(d*x + c)^2, x)
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(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)*sqrt(b*cos(d*x + c) + a) *cos(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(1/2)*(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))^(1/2)*(A + B*cos(c + d*x) + C*cos( c + d*x)^2), x)
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a \] Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(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)**4,x)*c + int(sqrt(cos(c + d*x)* b + a)*cos(c + d*x)**3,x)*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2 ,x)*a