Integrand size = 33, antiderivative size = 386 \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\frac {2 \left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {2 \left (36 a A b-24 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac {2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac {2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d} \]
-2/315*(36*A*a*b-24*B*a^2-49*B*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^3/ d+2/21*(3*A*b-2*B*a)*cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+2/ 9*B*cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d+2/315*(24*A*a^2*b+7 5*A*b^3-16*B*a^3-36*B*a*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^3/d+2/315 *(24*A*a^3*b+57*A*a*b^3-16*B*a^4-24*B*a^2*b^2+147*B*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^4/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/3 15*(a^2-b^2)*(24*A*a^2*b+75*A*b^3-16*B*a^3-36*B*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*cos(d*x+c))/(a+b))^(1/2)/b^4/d/(a+b*cos(d*x+c))^(1/2)
Time = 1.41 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.76 \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (6 a^2 A b+75 A b^3-4 a^3 B+111 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )-b (a+b \cos (c+d x)) \left (-2 \left (-48 a^2 A b+345 A b^3+32 a^3 B+57 a b^2 B\right ) \sin (c+d x)-b \left (\left (36 a A b-24 a^2 B+266 b^2 B\right ) \sin (2 (c+d x))+5 b (2 (9 A b+a B) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right )\right )}{1260 b^4 d \sqrt {a+b \cos (c+d x)}} \]
(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(6*a^2*A*b + 75*A*b^3 - 4*a^3*B + 111*a*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (24*a^3*A*b + 57*a *A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*((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*A*b + 345*A*b^3 + 32*a^3*B + 57*a*b^2*B)*Sin[c + d*x] - b*((36*a*A*b - 24*a^2*B + 266*b^2*B)*Sin[2*(c + d*x)] + 5*b*(2*(9 *A*b + a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260*b^4*d*Sqrt [a + b*Cos[c + d*x]])
Time = 2.18 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3469, 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.
| \(\displaystyle \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3469 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (3 (3 A b-2 a B) \cos ^2(c+d x)+7 b B \cos (c+d x)+4 a B\right )dx}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (3 (3 A b-2 a B) \cos ^2(c+d x)+7 b B \cos (c+d x)+4 a B\right )dx}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 (3 A b-2 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+7 b B \sin \left (c+d x+\frac {\pi }{2}\right )+4 a B\right )dx}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (-\left (\left (-24 B a^2+36 A b a-49 b^2 B\right ) \cos ^2(c+d x)\right )+b (45 A b-2 a B) \cos (c+d x)+6 a (3 A b-2 a B)\right )dx}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \sqrt {a+b \cos (c+d x)} \left (-\left (\left (-24 B a^2+36 A b a-49 b^2 B\right ) \cos ^2(c+d x)\right )+b (45 A b-2 a B) \cos (c+d x)+6 a (3 A b-2 a B)\right )dx}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (24 B a^2-36 A b a+49 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (45 A b-2 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+6 a (3 A b-2 a B)\right )dx}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-4 B a^2+6 A b a-49 b^2 B\right )-\left (-16 B a^3+24 A b a^2-36 b^2 B a+75 A b^3\right ) \cos (c+d x)\right )dx}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \sqrt {a+b \cos (c+d x)} \left (b \left (-4 B a^2+6 A b a-49 b^2 B\right )-\left (-16 B a^3+24 A b a^2-36 b^2 B a+75 A b^3\right ) \cos (c+d x)\right )dx}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-4 B a^2+6 A b a-49 b^2 B\right )+\left (16 B a^3-24 A b a^2+36 b^2 B a-75 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {2}{3} \int -\frac {b \left (-4 B a^3+6 A b a^2+111 b^2 B a+75 A b^3\right )+\left (-16 B a^4+24 A b a^3-24 b^2 B a^2+57 A b^3 a+147 b^4 B\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {1}{3} \int \frac {b \left (-4 B a^3+6 A b a^2+111 b^2 B a+75 A b^3\right )+\left (-16 B a^4+24 A b a^3-24 b^2 B a^2+57 A b^3 a+147 b^4 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (-\frac {1}{3} \int \frac {b \left (-4 B a^3+6 A b a^2+111 b^2 B a+75 A b^3\right )+\left (-16 B a^4+24 A b a^3-24 b^2 B a^2+57 A b^3 a+147 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {3 \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac {3 \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )}{5 b}}{7 b}+\frac {6 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}}{9 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}\) |
(2*B*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d) + ((6 *(3*A*b - 2*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7* b*d) + ((-2*(36*a*A*b - 24*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Si n[c + d*x])/(5*b*d) - (3*(((-2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^ 2*b^2*B + 147*b^4*B)*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)*(24*a ^2*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*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*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/(5*b))/(7*b))/(9*b)
3.3.96.3.1 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[(-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. \(1634\) vs. \(2(416)=832\).
Time = 17.28 (sec) , antiderivative size = 1635, normalized size of antiderivative = 4.24
| 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* B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^5+(720*A*b^5+640*B*a*b^4+2240 *B*b^5)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-432*A*a*b^4-1080*A*b^5+8 *B*a^2*b^3-960*B*a*b^4-2072*B*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c) +(-12*A*a^2*b^3+432*A*a*b^4+840*A*b^5+8*B*a^3*b^2-8*B*a^2*b^3+728*B*a*b^4+ 952*B*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(24*A*a^3*b^2+6*A*a^2*b ^3-258*A*a*b^4-240*A*b^5-16*B*a^4*b-4*B*a^3*b^2-24*B*a^2*b^3-204*B*a*b^4-1 68*B*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-24*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(co s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b-51*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^2*b^3+75*A*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))+24*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-24*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^3*b^2+57*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^2*b^3-57*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.66 \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\frac {\sqrt {2} {\left (-32 i \, B a^{5} + 48 i \, A a^{4} b - 36 i \, B a^{3} b^{2} + 96 i \, A a^{2} b^{3} - 39 i \, B a b^{4} - 225 i \, A 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 (32 i \, B a^{5} - 48 i \, A a^{4} b + 36 i \, B a^{3} b^{2} - 96 i \, A a^{2} b^{3} + 39 i \, B a b^{4} + 225 i \, A 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 (16 i \, B a^{4} b - 24 i \, A a^{3} b^{2} + 24 i \, B a^{2} b^{3} - 57 i \, A a b^{4} - 147 i \, B 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 (-16 i \, B a^{4} b + 24 i \, A a^{3} b^{2} - 24 i \, B a^{2} b^{3} + 57 i \, A a b^{4} + 147 i \, B 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 \, B b^{5} \cos \left (d x + c\right )^{3} + 8 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3} + 13 \, B a b^{4} + 75 \, A b^{5} + 5 \, {\left (B a b^{4} + 9 \, A b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (6 \, B a^{2} b^{3} - 9 \, A a b^{4} - 49 \, B 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^{5} d} \]
1/945*(sqrt(2)*(-32*I*B*a^5 + 48*I*A*a^4*b - 36*I*B*a^3*b^2 + 96*I*A*a^2*b ^3 - 39*I*B*a*b^4 - 225*I*A*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)*(32*I*B*a^5 - 48*I*A*a^4*b + 36*I*B*a^3*b^2 - 96*I*A*a^2*b^3 + 39*I*B*a*b^4 + 225*I*A*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(2)*(16*I*B*a^4*b - 24*I*A*a^3*b ^2 + 24*I*B*a^2*b^3 - 57*I*A*a*b^4 - 147*I*B*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(2)*(-16*I*B*a^4*b + 24*I*A*a^3* b^2 - 24*I*B*a^2*b^3 + 57*I*A*a*b^4 + 147*I*B*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*(35*B*b^5*cos(d*x + c)^3 + 8*B*a^3* b^2 - 12*A*a^2*b^3 + 13*B*a*b^4 + 75*A*b^5 + 5*(B*a*b^4 + 9*A*b^5)*cos(d*x + c)^2 - (6*B*a^2*b^3 - 9*A*a*b^4 - 49*B*b^5)*cos(d*x + c))*sqrt(b*cos(d* x + c) + a)*sin(d*x + c))/(b^5*d)
Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\text {Timed out} \]
\[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3} \,d x } \]
\[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \]