Integrand size = 41, antiderivative size = 408 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (18 a^3 b B-246 a b^3 B-8 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (21 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d} \] Output:
-2/315*(18*B*a^3*b-246*B*a*b^3-8*a^4*C-21*b^4*(9*A+7*C)-3*a^2*b^2*(21*A+11 *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)+2/315*(a^2-b^2)*(18*B*a^2*b-7 5*B*b^3-8*a^3*C-3*a*b^2*(21*A+13*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*Invers eJacobiAM(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)-2/315*(18*B*a^2*b-75*B*b^3-8*a^3*C-3*a*b^2*(21*A+13*C))*(a+b*cos(d*x+c ))^(1/2)*sin(d*x+c)/b^2/d+2/315*(63*A*b^2-18*B*a*b+8*C*a^2+49*C*b^2)*(a+b* cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+2/63*(9*B*b-4*C*a)*(a+b*cos(d*x+c))^(5/ 2)*sin(d*x+c)/b^2/d+2/9*C*cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d
Time = 3.02 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (153 a^2 b B+75 b^3 B+2 a^3 C+6 a b^2 (42 A+31 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-18 a^3 b B+246 a b^3 B+8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (72 a^2 b B+690 b^3 B-32 a^3 C+12 a b^2 (84 A+67 C)\right ) \sin (c+d x)+b \left (2 \left (126 A b^2+144 a b B+6 a^2 C+133 b^2 C\right ) \sin (2 (c+d x))+5 b (2 (9 b B+10 a C) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right )\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(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*(153*a^2*b*B + 75*b^3*B + 2*a^3 *C + 6*a*b^2*(42*A + 31*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-18*a ^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11 *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])*((72*a^2*b*B + 690*b^3*B - 32*a^3*C + 12*a*b^2*(84*A + 67*C))*Sin[c + d*x] + b*(2*(126*A*b^2 + 144*a* b*B + 6*a^2*C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + 10*a*C)*Sin[ 3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d *x]])
Time = 2.18 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.04, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left ((9 b B-4 a C) \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{3/2} \left ((9 b B-4 a C) \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((9 b B-4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+63 A b^2+49 b^2 C\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (a+b \cos (c+d x))^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+63 A b^2+49 b^2 C\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+63 A b^2+49 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-2 C a^2+57 b B a+63 A b^2+49 b^2 C\right )-\left (-8 C a^3+18 b B a^2-3 b^2 (21 A+13 C) a-75 b^3 B\right ) \cos (c+d x)\right )dx+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (-2 C a^2+57 b B a+63 A b^2+49 b^2 C\right )-\left (-8 C a^3+18 b B a^2-3 b^2 (21 A+13 C) a-75 b^3 B\right ) \cos (c+d x)\right )dx+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-2 C a^2+57 b B a+63 A b^2+49 b^2 C\right )+\left (8 C a^3-18 b B a^2+3 b^2 (21 A+13 C) a+75 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {2}{3} \int \frac {b \left (2 C a^3+153 b B a^2+6 b^2 (42 A+31 C) a+75 b^3 B\right )-\left (-8 C a^4+18 b B a^3-3 b^2 (21 A+11 C) a^2-246 b^3 B a-21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (2 C a^3+153 b B a^2+6 b^2 (42 A+31 C) a+75 b^3 B\right )-\left (-8 C a^4+18 b B a^3-3 b^2 (21 A+11 C) a^2-246 b^3 B a-21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (2 C a^3+153 b B a^2+6 b^2 (42 A+31 C) a+75 b^3 B\right )+\left (8 C a^4-18 b B a^3+3 b^2 (21 A+11 C) a^2+246 b^3 B a+21 b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \sin (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
Input:
Int[Cos[c + d*x]*(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]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d) + ((2*( 9*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d) + ((2*(63* A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (3*(((-2*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*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)*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*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*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*( 21*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/(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. \(2(389)=778\).
Time = 52.29 (sec) , antiderivative size = 2143, normalized size of antiderivative = 5.25
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2143\) |
| parts | \(\text {Expression too large to display}\) | \(2487\) |
Input:
int(cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,me thod=_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)*(-8*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+8*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)^1 0-246*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^4-63*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+63*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+18*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^2-18*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 = 0.13 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.71 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="fricas")
Output:
-2/945*(sqrt(1/2)*(-16*I*C*a^5 + 36*I*B*a^4*b - 6*I*(21*A + 10*C)*a^3*b^2 - 33*I*B*a^2*b^3 + 6*I*(63*A + 44*C)*a*b^4 + 225*I*B*b^5)*sqrt(b)*weierstr assPInverse(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)*(16*I*C*a^5 - 36* I*B*a^4*b + 6*I*(21*A + 10*C)*a^3*b^2 + 33*I*B*a^2*b^3 - 6*I*(63*A + 44*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)*(-8*I*C*a^4*b + 18*I*B*a^3*b^2 - 3*I*(21*A + 11*C)* a^2*b^3 - 246*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)*(8*I*C*a^4*b - 18*I*B*a^3*b^ 2 + 3*I*(21*A + 11*C)*a^2*b^3 + 246*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*(35*C*b^5*cos(d* x + c)^3 - 4*C*a^3*b^2 + 9*B*a^2*b^3 + 2*(63*A + 44*C)*a*b^4 + 75*B*b^5 + 5*(10*C*a*b^4 + 9*B*b^5)*cos(d*x + c)^2 + (3*C*a^2*b^3 + 72*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^4*d )
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)*(a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)* *2),x)
Output:
Timed out
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \] Input:
integrate(cos(d*x+c)*(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), x)
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \] Input:
integrate(cos(d*x+c)*(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), x)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int(cos(c + d*x)*(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)*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b^{2}+2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a b \] Input:
int(cos(d*x+c)*(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),x)*a**2 + int(sqrt(cos(c + d*x)* b + a)*cos(c + d*x)**4,x)*b*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)* *3,x)*a*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*b**2 + 2*int(s qrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a*b