Integrand size = 43, antiderivative size = 294 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac {2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d} \]
2/15015*a*(10439*A+9230*B+8368*C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/14 3*a*(13*B+5*C)*cos(d*x+c)^3*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/13*C*cos (d*x+c)^3*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d+2/6435*a^3*(10439*A+9230*B+8 368*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/9009*a^3*(2717*A+2522*B+2224* C)*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-4/45045*a^2*(10439*A+9 230*B+8368*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/1287*a^2*(143*A+182*B+ 136*C)*cos(d*x+c)^3*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d
Time = 1.53 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.61 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (3233516 A+2980640 B+2798182 C+4 (445588 A+454285 B+453146 C) \cos (c+d x)+(581152 A+676000 B+746519 C) \cos (2 (c+d x))+148720 A \cos (3 (c+d x))+225550 B \cos (3 (c+d x))+287060 C \cos (3 (c+d x))+20020 A \cos (4 (c+d x))+58240 B \cos (4 (c+d x))+94010 C \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+23940 C \cos (5 (c+d x))+3465 C \cos (6 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{720720 d} \]
Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3233516*A + 2980640*B + 2798182*C + 4*(44 5588*A + 454285*B + 453146*C)*Cos[c + d*x] + (581152*A + 676000*B + 746519 *C)*Cos[2*(c + d*x)] + 148720*A*Cos[3*(c + d*x)] + 225550*B*Cos[3*(c + d*x )] + 287060*C*Cos[3*(c + d*x)] + 20020*A*Cos[4*(c + d*x)] + 58240*B*Cos[4* (c + d*x)] + 94010*C*Cos[4*(c + d*x)] + 8190*B*Cos[5*(c + d*x)] + 23940*C* Cos[5*(c + d*x)] + 3465*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(720720*d)
Time = 1.83 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 27, 3042, 3460, 3042, 3238, 27, 3042, 3230, 3042, 3125}
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.
| \(\displaystyle \int \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos ^2(c+d x) (\cos (c+d x) a+a)^{5/2} (a (13 A+6 C)+a (13 B+5 C) \cos (c+d x))dx}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (\cos (c+d x) a+a)^{5/2} (a (13 A+6 C)+a (13 B+5 C) \cos (c+d x))dx}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (13 A+6 C)+a (13 B+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {1}{2} \cos ^2(c+d x) (\cos (c+d x) a+a)^{3/2} \left ((143 A+78 B+96 C) a^2+(143 A+182 B+136 C) \cos (c+d x) a^2\right )dx+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^{3/2} \left ((143 A+78 B+96 C) a^2+(143 A+182 B+136 C) \cos (c+d x) a^2\right )dx+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((143 A+78 B+96 C) a^2+(143 A+182 B+136 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^2(c+d x) \sqrt {\cos (c+d x) a+a} \left (3 (715 A+598 B+560 C) a^3+(2717 A+2522 B+2224 C) \cos (c+d x) a^3\right )dx+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \int \cos ^2(c+d x) \sqrt {\cos (c+d x) a+a} \left (3 (715 A+598 B+560 C) a^3+(2717 A+2522 B+2224 C) \cos (c+d x) a^3\right )dx+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (3 (715 A+598 B+560 C) a^3+(2717 A+2522 B+2224 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \int \cos ^2(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \left (\frac {2 \int \frac {1}{2} (3 a-2 a \cos (c+d x)) \sqrt {\cos (c+d x) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \left (\frac {\int (3 a-2 a \cos (c+d x)) \sqrt {\cos (c+d x) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \left (\frac {\int \left (3 a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \left (\frac {\frac {7}{3} a \int \sqrt {\cos (c+d x) a+a}dx-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \left (\frac {\frac {7}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )+\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}\right )+\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {2 a^2 (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {1}{11} \left (\frac {2 a^3 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{9 d}+\frac {1}{9} \left (\frac {2 a^4 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt {a \cos (c+d x)+a}}+\frac {3}{7} a^3 (10439 A+9230 B+8368 C) \left (\frac {\frac {14 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}-\frac {4 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}}{5 a}+\frac {2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}\right )\right )\right )}{13 a}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}\) |
(2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d) + ((2* a^2*(13*B + 5*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/( 11*d) + ((2*a^3*(143*A + 182*B + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + ((2*a^4*(2717*A + 2522*B + 2224*C)*Cos[c + d*x ]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a^3*(10439*A + 9230* B + 8368*C)*((2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d) + ((14*a^ 2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*Sqrt[a + a*Cos[c + d *x]]*Sin[c + d*x])/(3*d))/(5*a)))/7)/9)/11)/(13*a)
3.4.91.3.1 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*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[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[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-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*(b*(m + 1) - a*Si n[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && ! LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-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 - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -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/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 36.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (55440 C \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-32760 B -262080 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20020 A +140140 B +520520 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-77220 A -244530 B -566280 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (117117 A +225225 B +369369 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90090 A -120120 B -150150 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45045 A +45045 B +45045 C \right ) \sqrt {2}}{45045 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(176\) |
| parts | \(\frac {8 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (140 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+39 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+52 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+104\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {8 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (504 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-364 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+178 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+100 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200\right ) \sqrt {2}}{693 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {8 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (55440 \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-70560 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+41720 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3800 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4449 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5932 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11864\right ) \sqrt {2}}{45045 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(338\) |
int(cos(d*x+c)^2*(a+cos(d*x+c)*a)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_RETURNVERBOSE)
8/45045*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(55440*C*sin(1/2*d*x+1/2 *c)^12+(-32760*B-262080*C)*sin(1/2*d*x+1/2*c)^10+(20020*A+140140*B+520520* C)*sin(1/2*d*x+1/2*c)^8+(-77220*A-244530*B-566280*C)*sin(1/2*d*x+1/2*c)^6+ (117117*A+225225*B+369369*C)*sin(1/2*d*x+1/2*c)^4+(-90090*A-120120*B-15015 0*C)*sin(1/2*d*x+1/2*c)^2+45045*A+45045*B+45045*C)*2^(1/2)/(a*cos(1/2*d*x+ 1/2*c)^2)^(1/2)/d
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.59 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (3465 \, C a^{2} \cos \left (d x + c\right )^{6} + 315 \, {\left (13 \, B + 38 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 35 \, {\left (143 \, A + 416 \, B + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (3718 \, A + 4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 4 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
2/45045*(3465*C*a^2*cos(d*x + c)^6 + 315*(13*B + 38*C)*a^2*cos(d*x + c)^5 + 35*(143*A + 416*B + 523*C)*a^2*cos(d*x + c)^4 + 5*(3718*A + 4615*B + 418 4*C)*a^2*cos(d*x + c)^3 + 3*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c)^2 + 4*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c) + 8*(10439*A + 9230*B + 8368*C)*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.49 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.13 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {572 \, {\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 130 \, {\left (63 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (3465 \, \sqrt {2} a^{2} \sin \left (\frac {13}{2} \, d x + \frac {13}{2} \, c\right ) + 20475 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 70070 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 193050 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 459459 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1066065 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3783780 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{1441440 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="maxima")
1/1441440*(572*(35*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 225*sqrt(2)*a^2*sin( 7/2*d*x + 7/2*c) + 756*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 2100*sqrt(2)*a^2 *sin(3/2*d*x + 3/2*c) + 8190*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + 130*(63*sqrt(2)*a^2*sin(11/2*d*x + 11/2*c) + 385*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 1287*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 3465*sqrt(2)*a^2*sin(5/ 2*d*x + 5/2*c) + 8778*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 31878*sqrt(2)*a^2 *sin(1/2*d*x + 1/2*c))*B*sqrt(a) + (3465*sqrt(2)*a^2*sin(13/2*d*x + 13/2*c ) + 20475*sqrt(2)*a^2*sin(11/2*d*x + 11/2*c) + 70070*sqrt(2)*a^2*sin(9/2*d *x + 9/2*c) + 193050*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 459459*sqrt(2)*a^2 *sin(5/2*d*x + 5/2*c) + 1066065*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 3783780 *sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d
Time = 11.59 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.30 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (3465 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {13}{2} \, d x + \frac {13}{2} \, c\right ) + 4095 \, {\left (2 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 10010 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 12870 \, {\left (10 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 15 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 9009 \, {\left (48 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 50 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 51 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 15015 \, {\left (80 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 76 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 71 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 180180 \, {\left (26 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 23 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 21 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{1441440 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="giac")
1/1441440*sqrt(2)*(3465*C*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(13/2*d*x + 13/ 2*c) + 4095*(2*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(11/2*d*x + 11/2*c) + 10010*(2*A*a^2*sgn(cos(1/2*d*x + 1/2*c) ) + 5*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 7*C*a^2*sgn(cos(1/2*d*x + 1/2*c))) *sin(9/2*d*x + 9/2*c) + 12870*(10*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 13*B*a ^2*sgn(cos(1/2*d*x + 1/2*c)) + 15*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(7/2 *d*x + 7/2*c) + 9009*(48*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 50*B*a^2*sgn(co s(1/2*d*x + 1/2*c)) + 51*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(5/2*d*x + 5/ 2*c) + 15015*(80*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 76*B*a^2*sgn(cos(1/2*d* x + 1/2*c)) + 71*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2*d*x + 3/2*c) + 1 80180*(26*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 23*B*a^2*sgn(cos(1/2*d*x + 1/2 *c)) + 21*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/2*d*x + 1/2*c))*sqrt(a)/d
Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]