Integrand size = 23, antiderivative size = 308 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (5 a^2-57 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b d}-\frac {2 \left (10 a^2-49 b^2\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b d}-\frac {4 a (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b d}+\frac {2 (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b d} \] Output:
-2/315*(10*a^4-279*a^2*b^2-147*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^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2 )+4/315*a*(5*a^4-62*a^2*b^2+57*b^4)*((a+b*cos(d*x+c))/(a+b))^(1/2)*Inverse JacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b^2/d/(a+b*cos(d*x+c))^(1/ 2)-4/315*a*(5*a^2-57*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d-2/315*(10* a^2-49*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d-4/63*a*(a+b*cos(d*x+c))^ (5/2)*sin(d*x+c)/b/d+2/9*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b/d
Time = 1.70 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.85 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\frac {-8 \left (10 a^5+10 a^4 b-279 a^3 b^2-279 a^2 b^3-147 a b^4-147 b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+16 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+b \left (40 a^4+1984 a^2 b^2+301 b^4+4 a b \left (160 a^2+619 b^2\right ) \cos (c+d x)+8 \left (85 a^2 b^2+42 b^4\right ) \cos (2 (c+d x))+260 a b^3 \cos (3 (c+d x))+35 b^4 \cos (4 (c+d x))\right ) \sin (c+d x)}{1260 b^2 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2),x]
Output:
(-8*(10*a^5 + 10*a^4*b - 279*a^3*b^2 - 279*a^2*b^3 - 147*a*b^4 - 147*b^5)* Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 16*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ell ipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(40*a^4 + 1984*a^2*b^2 + 301*b^4 + 4*a*b*(160*a^2 + 619*b^2)*Cos[c + d*x] + 8*(85*a^2*b^2 + 42*b^4)*Cos[2*(c + d*x)] + 260*a*b^3*Cos[3*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)])*Sin[c + d* x])/(1260*b^2*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.72 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 3270, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (7 b-2 a \cos (c+d x)) (a+b \cos (c+d x))^{5/2}dx}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (7 b-2 a \cos (c+d x)) (a+b \cos (c+d x))^{5/2}dx}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (7 b-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (39 a b-\left (10 a^2-49 b^2\right ) \cos (c+d x)\right )dx-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (39 a b-\left (10 a^2-49 b^2\right ) \cos (c+d x)\right )dx-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (39 a b+\left (49 b^2-10 a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (55 a^2+49 b^2\right )-2 a \left (5 a^2-57 b^2\right ) \cos (c+d x)\right )dx-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (55 a^2+49 b^2\right )-2 a \left (5 a^2-57 b^2\right ) \cos (c+d x)\right )dx-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (55 a^2+49 b^2\right )-2 a \left (5 a^2-57 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {a b \left (155 a^2+261 b^2\right )-\left (10 a^4-279 b^2 a^2-147 b^4\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {a b \left (155 a^2+261 b^2\right )-\left (10 a^4-279 b^2 a^2-147 b^4\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {a b \left (155 a^2+261 b^2\right )+\left (-10 a^4+279 b^2 a^2+147 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (10 a^4-279 a^2 b^2-147 b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (10 a^4-279 a^2 b^2-147 b^4\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {4 a \left (5 a^4-62 a^2 b^2+57 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (10 a^4-279 a^2 b^2-147 b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (5 a^2-57 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (10 a^2-49 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}\) |
Input:
Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2),x]
Output:
(2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d) + ((-4*a*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((-2*(10*a^2 - 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (3*(((-2*(10*a^4 - 279*a^2*b^2 - 147*b^4 )*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqr t[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*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 - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Cos[c + d *x]]*Sin[c + d*x])/(3*d)))/5)/7)/(9*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[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[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Ne Q[a^2 - b^2, 0] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(994\) vs. \(2(289)=578\).
Time = 13.03 (sec) , antiderivative size = 995, normalized size of antiderivative = 3.23
Input:
int(cos(d*x+c)^2*(a+cos(d*x+c)*b)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/315*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^5+(2080*a*b^4+2240*b^5)*sin(1/2 *d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1360*a^2*b^3-3120*a*b^4-2072*b^5)*sin(1 /2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(320*a^3*b^2+1360*a^2*b^3+2408*a*b^4+95 2*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-10*a^4*b-160*a^3*b^2-666* a^2*b^3-684*a*b^4-168*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+10*(sin (1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/ 2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5-124*(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)*Ellipti cF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^2+114*(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*b^4-10*(sin(1/2*d*x+1/2*c)^2)^(1/2)* (-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+ 1/2*c),(-2*b/(a-b))^(1/2))*a^5+10*(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+279*(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^2-279*(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+147*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.67 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Output:
-2/945*(sqrt(1/2)*(20*I*a^5 - 93*I*a^3*b^2 + 489*I*a*b^4)*sqrt(b)*weierstr assPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b *cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(-20*I*a^5 + 93*I *a^3*b^2 - 489*I*a*b^4)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^ 2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(10*I*a^4*b - 279*I*a^2*b^3 - 147*I*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, weie rstrassPInverse(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)*(-10*I*a^4 *b + 279*I*a^2*b^3 + 147*I*b^5)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2 )/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2 )/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(35*b^5*cos(d*x + c)^3 + 95*a*b^4*cos(d*x + c)^2 + 5*a ^3*b^2 + 163*a*b^4 + (75*a^2*b^3 + 49*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^3*d)
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(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)^2, x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(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)^2, x)
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2),x)
Output:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2), x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}d x \right ) b^{2}+2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2),x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4,x)*b**2 + 2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*a*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a**2