Integrand size = 40, antiderivative size = 378 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (18 a^3 b B-246 a b^3 B-8 a^4 C-33 a^2 b^2 C-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-39 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a b B-8 a^2 C-49 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d} \]
-2/315*(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-2/315*(18*B*a^2*b-75*B*b^3-8*C*a^ 3-39*C*a*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d-2/315*(18*B*a^3*b-24 6*B*a*b^3-8*C*a^4-33*C*a^2*b^2-147*C*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos (1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b *cos(d*x+c))^(1/2)/b^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/315*(a^2-b^2)*(1 8*B*a^2*b-75*B*b^3-8*C*a^3-39*C*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/ 2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*((a+b*c os(d*x+c))/(a+b))^(1/2)/b^3/d/(a+b*cos(d*x+c))^(1/2)
Time = 2.78 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.77 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (153 a^2 b B+75 b^3 B+2 a^3 C+186 a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-18 a^3 b B+246 a b^3 B+8 a^4 C+33 a^2 b^2 C+147 b^4 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (72 a^2 b B+690 b^3 B-32 a^3 C+804 a b^2 C\right ) \sin (c+d x)+b \left (2 \left (144 a b B+6 a^2 C+133 b^2 C\right ) \sin (2 (c+d x))+5 b (2 (9 b B+10 a C) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right )\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \]
(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 + 186*a*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-18*a^3*b*B + 2 46*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*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 + 804*a*b^2*C)*Sin[c + d*x] + 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 = 2.16 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.04, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.575, Rules used = {3042, 3508, 3042, 3469, 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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (B+C \cos (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3469 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left ((9 b B-4 a C) \cos ^2(c+d x)+7 b C \cos (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{3/2} \left ((9 b B-4 a C) \cos ^2(c+d x)+7 b C \cos (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((9 b B-4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+7 b C \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b (15 b B-2 a C)-\left (-8 C a^2+18 b B a-49 b^2 C\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (a+b \cos (c+d x))^{3/2} \left (3 b (15 b B-2 a C)-\left (-8 C a^2+18 b B a-49 b^2 C\right ) \cos (c+d x)\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+49 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-2 C a^2+57 b B a+49 b^2 C\right )-\left (-8 C a^3+18 b B a^2-39 b^2 C a-75 b^3 B\right ) \cos (c+d x)\right )dx-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (-2 C a^2+57 b B a+49 b^2 C\right )-\left (-8 C a^3+18 b B a^2-39 b^2 C a-75 b^3 B\right ) \cos (c+d x)\right )dx-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-2 C a^2+57 b B a+49 b^2 C\right )+\left (8 C a^3-18 b B a^2+39 b^2 C a+75 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {2}{3} \int \frac {b \left (2 C a^3+153 b B a^2+186 b^2 C a+75 b^3 B\right )-\left (-8 C a^4+18 b B a^3-33 b^2 C a^2-246 b^3 B a-147 b^4 C\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (2 C a^3+153 b B a^2+186 b^2 C a+75 b^3 B\right )-\left (-8 C a^4+18 b B a^3-33 b^2 C a^2-246 b^3 B a-147 b^4 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (2 C a^3+153 b B a^2+186 b^2 C a+75 b^3 B\right )+\left (8 C a^4-18 b B a^3+33 b^2 C a^2+246 b^3 B a+147 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (-8 a^2 C+18 a b B-49 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
(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*(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 - 33*a^2*b^2*C - 147*b^4* C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sq rt[(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 - 39*a*b^2*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 - 39*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x]) /(3*d)))/5)/(7*b))/(9*b)
3.9.20.3.1 Defintions of rubi rules used
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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^( n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*( m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin [e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !(IGt Q[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1634\) vs. \(2(408)=816\).
Time = 18.18 (sec) , antiderivative size = 1635, normalized size of antiderivative = 4.33
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1635\) |
| parts | \(\text {Expression too large to display}\) | \(1824\) |
-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)*(-1120* C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^5+(720*B*b^5+1360*C*a*b^4+224 0*C*b^5)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-936*B*a*b^4-1080*B*b^5- 424*C*a^2*b^3-2040*C*a*b^4-2072*C*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/ 2*c)+(324*B*a^2*b^3+936*B*a*b^4+840*B*b^5-4*C*a^3*b^2+424*C*a^2*b^3+1568*C *a*b^4+952*C*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-18*B*a^3*b^2-1 62*B*a^2*b^3-384*B*a*b^4-240*B*b^5+8*C*a^4*b+2*C*a^3*b^2-282*C*a^2*b^3-444 *C*a*b^4-168*C*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+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)*El lipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b-93*B*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)*Ellip ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+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))-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)*EllipticE(cos(1/2*d*x+1 /2*c),(-2*b/(a-b))^(1/2))*a^4*b+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)*EllipticE(cos(1/2*d*x+1/2*c),( -2*b/(a-b))^(1/2))*a^3*b^2+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^2*b^3-246*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*s...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.69 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (16 i \, C a^{5} - 36 i \, B a^{4} b + 60 i \, C a^{3} b^{2} + 33 i \, B a^{2} b^{3} - 264 i \, C a b^{4} - 225 i \, B b^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-16 i \, C a^{5} + 36 i \, B a^{4} b - 60 i \, C a^{3} b^{2} - 33 i \, B a^{2} b^{3} + 264 i \, C a b^{4} + 225 i \, B b^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-8 i \, C a^{4} b + 18 i \, B a^{3} b^{2} - 33 i \, C a^{2} b^{3} - 246 i \, B a b^{4} - 147 i \, C b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (8 i \, C a^{4} b - 18 i \, B a^{3} b^{2} + 33 i \, C a^{2} b^{3} + 246 i \, B a b^{4} + 147 i \, C b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (35 \, C b^{5} \cos \left (d x + c\right )^{3} - 4 \, C a^{3} b^{2} + 9 \, B a^{2} b^{3} + 88 \, C a b^{4} + 75 \, B b^{5} + 5 \, {\left (10 \, C a b^{4} + 9 \, B b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, C a^{2} b^{3} + 72 \, B a b^{4} + 49 \, C b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{945 \, b^{4} d} \]
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="fricas")
1/945*(sqrt(2)*(16*I*C*a^5 - 36*I*B*a^4*b + 60*I*C*a^3*b^2 + 33*I*B*a^2*b^ 3 - 264*I*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) + sqrt(2)*(-16*I*C*a^5 + 36*I*B*a^4*b - 60*I*C*a^3*b^2 - 33*I*B*a^2*b^3 + 264*I*C*a*b^4 + 225*I*B*b^5)*sqrt(b)*weierstrassPInver se(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(2)*(-8*I*C*a^4*b + 18*I*B*a^3 *b^2 - 33*I*C*a^2*b^3 - 246*I*B*a*b^4 - 147*I*C*b^5)*sqrt(b)*weierstrassZe ta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInver se(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(2)*(8*I*C*a^4*b - 18*I*B*a^3 *b^2 + 33*I*C*a^2*b^3 + 246*I*B*a*b^4 + 147*I*C*b^5)*sqrt(b)*weierstrassZe ta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInver se(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*(35*C*b^5*cos(d*x + c)^3 - 4*C*a^ 3*b^2 + 9*B*a^2*b^3 + 88*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 + 49*C*b^5)*cos(d*x + c))*sqrt(b*c os(d*x + c) + a)*sin(d*x + c))/(b^4*d)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="maxima")
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="giac")
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]