Integrand size = 23, antiderivative size = 371 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\frac {2 a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac {2 a \left (8 a^2+67 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2+81 b^2\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \] Output:
2/693*a*(8*a^4+51*a^2*b^2+741*b^4)*(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) -2/693*(8*a^6+49*a^4*b^2+78*a^2*b^4-135*b^6)*((a+b*cos(d*x+c))/(a+b))^(1/2 )*InverseJacobiAM(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)+2/693*(8*a^4+57*a^2*b^2+135*b^4)*(a+b*cos(d*x+c))^(1/2)*sin(d* x+c)/b^2/d+2/693*a*(8*a^2+67*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+ 2/693*(8*a^2+81*b^2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^2/d-8/99*a*(a+b*c os(d*x+c))^(7/2)*sin(d*x+c)/b^2/d+2/11*cos(d*x+c)*(a+b*cos(d*x+c))^(7/2)*s in(d*x+c)/b/d
Time = 1.60 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.72 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (2 a^4 b+663 a^2 b^3+135 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )-b (a+b \cos (c+d x)) \left (\left (64 a^4-3732 a^2 b^2-2610 b^4\right ) \sin (c+d x)-b \left (4 \left (6 a^3+619 a b^2\right ) \sin (2 (c+d x))+b \left (\left (452 a^2+513 b^2\right ) \sin (3 (c+d x))+7 b (46 a \sin (4 (c+d x))+9 b \sin (5 (c+d x)))\right )\right )\right )}{5544 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2),x]
Output:
(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^4*b + 663*a^2*b^3 + 135*b^5 )*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(8*a^4 + 51*a^2*b^2 + 741*b^4) *((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])*((64*a^4 - 3732*a^2*b^2 - 2610* b^4)*Sin[c + d*x] - b*(4*(6*a^3 + 619*a*b^2)*Sin[2*(c + d*x)] + b*((452*a^ 2 + 513*b^2)*Sin[3*(c + d*x)] + 7*b*(46*a*Sin[4*(c + d*x)] + 9*b*Sin[5*(c + d*x)])))))/(5544*b^3*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.17 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.043, Rules used = {3042, 3272, 27, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (-4 a \cos ^2(c+d x)+9 b \cos (c+d x)+2 a\right )dx}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{5/2} \left (-4 a \cos ^2(c+d x)+9 b \cos (c+d x)+2 a\right )dx}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (-4 a \sin \left (c+d x+\frac {\pi }{2}\right )^2+9 b \sin \left (c+d x+\frac {\pi }{2}\right )+2 a\right )dx}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int -\frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (10 a b-\left (8 a^2+81 b^2\right ) \cos (c+d x)\right )dx}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int (a+b \cos (c+d x))^{5/2} \left (10 a b-\left (8 a^2+81 b^2\right ) \cos (c+d x)\right )dx}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (10 a b+\left (-8 a^2-81 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {\frac {2}{7} \int \frac {5}{2} (a+b \cos (c+d x))^{3/2} \left (3 b \left (2 a^2-27 b^2\right )-a \left (8 a^2+67 b^2\right ) \cos (c+d x)\right )dx-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \int (a+b \cos (c+d x))^{3/2} \left (3 b \left (2 a^2-27 b^2\right )-a \left (8 a^2+67 b^2\right ) \cos (c+d x)\right )dx-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (2 a^2-27 b^2\right )-a \left (8 a^2+67 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (2 a b \left (a^2-101 b^2\right )-\left (8 a^4+57 b^2 a^2+135 b^4\right ) \cos (c+d x)\right )dx-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (2 a b \left (a^2-101 b^2\right )-\left (8 a^4+57 b^2 a^2+135 b^4\right ) \cos (c+d x)\right )dx-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (2 a b \left (a^2-101 b^2\right )+\left (-8 a^4-57 b^2 a^2-135 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int -\frac {b \left (2 a^4+663 b^2 a^2+135 b^4\right )+a \left (8 a^4+51 b^2 a^2+741 b^4\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (2 a^4+663 b^2 a^2+135 b^4\right )+a \left (8 a^4+51 b^2 a^2+741 b^4\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (2 a^4+663 b^2 a^2+135 b^4\right )+a \left (8 a^4+51 b^2 a^2+741 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {\frac {5}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^4+51 a^2 b^2+741 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4+57 a^2 b^2+135 b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 a \left (8 a^2+67 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2+81 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2),x]
Output:
(2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d) + ((-8*a *(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d) - ((-2*(8*a^2 + 81*b^2)* (a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (5*((-2*a*(8*a^2 + 67*b^2 )*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (3*(((-2*a*(8*a^4 + 51* a^2*b^2 + 741*b^4)*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*(8*a^6 + 49*a^4*b^2 + 78*a^2*b^4 - 135*b^6)*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*(8*a^4 + 5 7*a^2*b^2 + 135*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5))/7) /(9*b))/(11*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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !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]
Leaf count of result is larger than twice the leaf count of optimal. \(1139\) vs. \(2(348)=696\).
Time = 18.79 (sec) , antiderivative size = 1140, normalized size of antiderivative = 3.07
Input:
int(cos(d*x+c)^3*(a+cos(d*x+c)*b)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/693*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4032*c os(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12*b^6+(-7168*a*b^5-10080*b^6)*sin(1/ 2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(4384*a^2*b^4+14336*a*b^5+11376*b^6)*si n(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-928*a^3*b^3-6576*a^2*b^4-13232*a*b ^5-6984*b^6)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(-4*a^4*b^2+928*a^3*b ^3+5024*a^2*b^4+6064*a*b^5+2772*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2* c)+(8*a^5*b+2*a^4*b^2-642*a^3*b^3-1416*a^2*b^4-1338*a*b^5-558*b^6)*sin(1/2 *d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-8*(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^6-49*(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^2-78*(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^4+135*(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))*b^6+8*( 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^6-8*(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^5*b+51*(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(co...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.51 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Output:
-2/2079*(sqrt(1/2)*(-16*I*a^6 - 96*I*a^4*b^2 + 507*I*a^2*b^4 + 405*I*b^6)* 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(1/2)*( 16*I*a^6 + 96*I*a^4*b^2 - 507*I*a^2*b^4 - 405*I*b^6)*sqrt(b)*weierstrassPI nverse(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*a^5*b - 51*I*a ^3*b^3 - 741*I*a*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*a^5*b + 51*I*a^3*b^3 + 741*I*a*b^5)*sqrt(b)*weier strassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstra ssPInverse(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*(63*b^6*cos(d*x + c)^4 + 161*a*b^5*cos(d*x + c)^3 - 4*a^4*b^2 + 205*a^2*b^4 + 135*b^6 + (113*a^2*b^ 4 + 81*b^6)*cos(d*x + c)^2 + (3*a^3*b^3 + 229*a*b^5)*cos(d*x + c))*sqrt(b* cos(d*x + c) + a)*sin(d*x + c))/(b^4*d)
Timed out. \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(a+b*cos(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^3, x)
\[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^3, x)
Timed out. \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int(cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2),x)
Output:
int(cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2), x)
\[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}d x \right ) b^{2}+2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^3*(a+b*cos(d*x+c))^(5/2),x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**5,x)*b**2 + 2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4,x)*a*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*a**2