Integrand size = 31, antiderivative size = 372 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {2 \left (45 a^3 A b+435 a A b^3-10 a^4 B+279 a^2 b^2 B+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (45 a^2 A b+75 A b^3-10 a^3 B+114 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b d}+\frac {2 \left (45 a A b-10 a^2 B+49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}+\frac {2 (9 A b-2 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 B (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d} \]
2/315*(45*A*a*b-10*B*a^2+49*B*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d+2 /63*(9*A*b-2*B*a)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d+2/9*B*(a+b*cos(d*x +c))^(7/2)*sin(d*x+c)/b/d+2/315*(45*A*a^2*b+75*A*b^3-10*B*a^3+114*B*a*b^2) *sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b/d+2/315*(45*A*a^3*b+435*A*a*b^3-10*B* a^4+279*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^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/315*(a^2-b^2)*(45*A*a^2*b+75* A*b^3-10*B*a^3+114*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^2/d/(a+b*cos(d*x+c))^(1/2)
Time = 3.98 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (405 a^2 A b+75 A b^3+155 a^3 B+261 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (45 a^3 A b+435 a A b^3-10 a^4 B+279 a^2 b^2 B+147 b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (2 \left (540 a^2 A b+345 A b^3+20 a^3 B+747 a b^2 B\right ) \sin (c+d x)+b \left (\left (540 a A b+300 a^2 B+266 b^2 B\right ) \sin (2 (c+d x))+5 b (2 (9 A b+19 a B) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right )\right )}{1260 b^2 d \sqrt {a+b \cos (c+d x)}} \]
(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(405*a^2*A*b + 75*A*b^3 + 155*a ^3*B + 261*a*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*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*(540*a^2*A*b + 345*A*b^3 + 20*a^3*B + 747*a*b^2*B)* Sin[c + d*x] + b*((540*a*A*b + 300*a^2*B + 266*b^2*B)*Sin[2*(c + d*x)] + 5 *b*(2*(9*A*b + 19*a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260 *b^2*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.98 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.03, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.742, Rules used = {3042, 3447, 3042, 3502, 27, 3042, 3232, 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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \int (a+b \cos (c+d x))^{5/2} \left (A \cos (c+d x)+B \cos ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A \sin \left (c+d x+\frac {\pi }{2}\right )+B \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} (7 b B+(9 A b-2 a B) \cos (c+d x))dx}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{5/2} (7 b B+(9 A b-2 a B) \cos (c+d x))dx}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (7 b B+(9 A b-2 a B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b (15 A b+13 a B)+\left (-10 B a^2+45 A b a+49 b^2 B\right ) \cos (c+d x)\right )dx+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (3 b (15 A b+13 a B)+\left (-10 B a^2+45 A b a+49 b^2 B\right ) \cos (c+d x)\right )dx+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b (15 A b+13 a B)+\left (-10 B a^2+45 A b a+49 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (55 B a^2+120 A b a+49 b^2 B\right )+\left (-10 B a^3+45 A b a^2+114 b^2 B a+75 A b^3\right ) \cos (c+d x)\right )dx+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (55 B a^2+120 A b a+49 b^2 B\right )+\left (-10 B a^3+45 A b a^2+114 b^2 B a+75 A b^3\right ) \cos (c+d x)\right )dx+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (55 B a^2+120 A b a+49 b^2 B\right )+\left (-10 B a^3+45 A b a^2+114 b^2 B a+75 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {b \left (155 B a^3+405 A b a^2+261 b^2 B a+75 A b^3\right )+\left (-10 B a^4+45 A b a^3+279 b^2 B a^2+435 A b^3 a+147 b^4 B\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (155 B a^3+405 A b a^2+261 b^2 B a+75 A b^3\right )+\left (-10 B a^4+45 A b a^3+279 b^2 B a^2+435 A b^3 a+147 b^4 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {b \left (155 B a^3+405 A b a^2+261 b^2 B a+75 A b^3\right )+\left (-10 B a^4+45 A b a^3+279 b^2 B a^2+435 A b^3 a+147 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 \left (-10 a^2 B+45 a A b+49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac {3}{5} \left (\frac {2 \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}\right )\right )\right )+\frac {2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
(2*B*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d) + ((2*(9*A*b - 2*a*B )*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((2*(45*a*A*b - 10*a^2* B + 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (3*(((2*(45 *a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*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)*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 1 14*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*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5 )/7)/(9*b)
3.4.12.3.1 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_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1634\) vs. \(2(402)=804\).
Time = 19.43 (sec) , antiderivative size = 1635, normalized size of antiderivative = 4.40
| 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+2080*B*a*b^4+224 0*B*b^5)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1440*A*a*b^4-1080*A*b^5 -1360*B*a^2*b^3-3120*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)+(1080*A*a^2*b^3+1440*A*a*b^4+840*A*b^5+320*B*a^3*b^2+1360*B*a^2*b^3 +2408*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)+(-270*A*a ^3*b^2-540*A*a^2*b^3-510*A*a*b^4-240*A*b^5-10*B*a^4*b-160*B*a^3*b^2-666*B* a^2*b^3-684*B*a*b^4-168*B*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+45* 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-45*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+435*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)*El lipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^3-435*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)*Ellipt icE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^4-45*A*(sin(1/2*d*x+1/2*c)^ 2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos (1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b-30*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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.72 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {\sqrt {2} {\left (-20 i \, B a^{5} + 90 i \, A a^{4} b + 93 i \, B a^{3} b^{2} - 345 i \, A a^{2} b^{3} - 489 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 (20 i \, B a^{5} - 90 i \, A a^{4} b - 93 i \, B a^{3} b^{2} + 345 i \, A a^{2} b^{3} + 489 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 (10 i \, B a^{4} b - 45 i \, A a^{3} b^{2} - 279 i \, B a^{2} b^{3} - 435 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 (-10 i \, B a^{4} b + 45 i \, A a^{3} b^{2} + 279 i \, B a^{2} b^{3} + 435 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} + 5 \, B a^{3} b^{2} + 135 \, A a^{2} b^{3} + 163 \, B a b^{4} + 75 \, A b^{5} + 5 \, {\left (19 \, B a b^{4} + 9 \, A b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, B a^{2} b^{3} + 135 \, 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^{3} d} \]
1/945*(sqrt(2)*(-20*I*B*a^5 + 90*I*A*a^4*b + 93*I*B*a^3*b^2 - 345*I*A*a^2* b^3 - 489*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*s in(d*x + c) + 2*a)/b) + sqrt(2)*(20*I*B*a^5 - 90*I*A*a^4*b - 93*I*B*a^3*b^ 2 + 345*I*A*a^2*b^3 + 489*I*B*a*b^4 + 225*I*A*b^5)*sqrt(b)*weierstrassPInv erse(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)*(10*I*B*a^4*b - 45*I*A*a ^3*b^2 - 279*I*B*a^2*b^3 - 435*I*A*a*b^4 - 147*I*B*b^5)*sqrt(b)*weierstras sZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPIn verse(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)*(-10*I*B*a^4*b + 45*I* A*a^3*b^2 + 279*I*B*a^2*b^3 + 435*I*A*a*b^4 + 147*I*B*b^5)*sqrt(b)*weierst rassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrass PInverse(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*co s(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) + 6*(35*B*b^5*cos(d*x + c)^3 + 5*B*a^3*b^2 + 135*A*a^2*b^3 + 163*B*a*b^4 + 75*A*b^5 + 5*(19*B*a*b^4 + 9*A *b^5)*cos(d*x + c)^2 + (75*B*a^2*b^3 + 135*A*a*b^4 + 49*B*b^5)*cos(d*x + c ))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^3*d)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\int \cos \left (c+d\,x\right )\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]