Integrand size = 43, antiderivative size = 302 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a (4 A b+9 a B) \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (4 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sin (c+d x)}{45 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (10 a A b+5 a^2 B+7 b^2 B+14 a b C\right ) \sin (c+d x)}{21 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \] Output:
-2/15*(18*B*a*b+3*b^2*(3*A+5*C)+a^2*(7*A+9*C))*EllipticE(sin(1/2*d*x+1/2*c ),2^(1/2))/d+2/21*(10*A*a*b+5*B*a^2+7*B*b^2+14*C*a*b)*InverseJacobiAM(1/2* d*x+1/2*c,2^(1/2))/d+2/63*a*(4*A*b+9*B*a)*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/ 45*(4*A*b^2+18*B*a*b+a^2*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/21*(10 *A*a*b+5*B*a^2+7*B*b^2+14*C*a*b)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/15*(18*B* a*b+3*b^2*(3*A+5*C)+a^2*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/9*A*(a+ b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(9/2)
Time = 8.78 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (-21 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+15 \left (5 a^2 B+7 b^2 B+2 a b (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {35 a^2 A \sin (c+d x)}{\cos ^{\frac {9}{2}}(c+d x)}+\frac {45 a (2 A b+a B) \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}+\frac {7 \left (9 A b^2+18 a b B+a^2 (7 A+9 C)\right ) \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {15 \left (5 a^2 B+7 b^2 B+2 a b (5 A+7 C)\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {21 \left (18 a b B+3 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{315 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(11/2),x]
Output:
(2*(-21*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d* x)/2, 2] + 15*(5*a^2*B + 7*b^2*B + 2*a*b*(5*A + 7*C))*EllipticF[(c + d*x)/ 2, 2] + (35*a^2*A*Sin[c + d*x])/Cos[c + d*x]^(9/2) + (45*a*(2*A*b + a*B)*S in[c + d*x])/Cos[c + d*x]^(7/2) + (7*(9*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C) )*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (15*(5*a^2*B + 7*b^2*B + 2*a*b*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(18*a*b*B + 3*b^2*(3*A + 5*C ) + a^2*(7*A + 9*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(315*d)
Time = 1.39 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.372, Rules used = {3042, 3526, 27, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 3116, 3042, 3119, 3120}
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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x)) \left (3 b (A+3 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+4 A b+9 a B\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x)) \left (3 b (A+3 C) \cos ^2(c+d x)+(7 a A+9 b B+9 a C) \cos (c+d x)+4 A b+9 a B\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 b (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 A b+9 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int -\frac {21 b^2 (A+3 C) \cos ^2(c+d x)+9 \left (5 B a^2+10 A b a+14 b C a+7 b^2 B\right ) \cos (c+d x)+7 \left ((7 A+9 C) a^2+18 b B a+4 A b^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 (A+3 C) \cos ^2(c+d x)+9 \left (5 B a^2+10 A b a+14 b C a+7 b^2 B\right ) \cos (c+d x)+7 \left ((7 A+9 C) a^2+18 b B a+4 A b^2\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (5 B a^2+10 A b a+14 b C a+7 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+7 \left ((7 A+9 C) a^2+18 b B a+4 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3 \left (15 \left (5 B a^2+10 A b a+14 b C a+7 b^2 B\right )+7 \left ((7 A+9 C) a^2+18 b B a+3 b^2 (3 A+5 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \frac {15 \left (5 B a^2+10 A b a+14 b C a+7 b^2 B\right )+7 \left ((7 A+9 C) a^2+18 b B a+3 b^2 (3 A+5 C)\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \frac {15 \left (5 B a^2+10 A b a+14 b C a+7 b^2 B\right )+7 \left ((7 A+9 C) a^2+18 b B a+3 b^2 (3 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)}dx+7 \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+7 \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {14 \sin (c+d x) \left (a^2 (7 A+9 C)+18 a b B+4 A b^2\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {3}{5} \left (15 \left (5 a^2 B+10 a A b+14 a b C+7 b^2 B\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+7 \left (a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {2 a (9 a B+4 A b) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*a *(4*A*b + 9*a*B)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((14*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (3*(1 5*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*((2*EllipticF[(c + d*x)/2, 2]) /(3*d) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))) + 7*(18*a*b*B + 3*b^2* (3*A + 5*C) + a^2*(7*A + 9*C))*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Sin[ c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1168\) vs. \(2(281)=562\).
Time = 8.11 (sec) , antiderivative size = 1169, normalized size of antiderivative = 3.87
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1169\) |
| parts | \(\text {Expression too large to display}\) | \(1422\) |
Input:
int((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x ,method=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/5*(A*b^2+2*B *a*b+C*a^2)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+ 1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c) ^6-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipt icE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c) *sin(1/2*d*x+1/2*c)^4+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2* sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*(-2*s in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*A*a^2*(-1/144*cos(1/2*d* x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x +1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 *d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c) ^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^ (1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/ (-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x +1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^ 2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF (cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+2*C*b ^2/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*...
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{2} + 2 i \, {\left (5 \, A + 7 \, C\right )} a b + 7 i \, B b^{2}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{2} - 2 i \, {\left (5 \, A + 7 \, C\right )} a b - 7 i \, B b^{2}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{2} + 18 i \, B a b + 3 i \, {\left (3 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{2} - 18 i \, B a b - 3 i \, {\left (3 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (21 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{2} + 18 \, B a b + 3 \, {\left (3 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (5 \, B a^{2} + 2 \, {\left (5 \, A + 7 \, C\right )} a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 35 \, A a^{2} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{2} + 18 \, B a b + 9 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 1/2),x, algorithm="fricas")
Output:
-1/315*(15*sqrt(2)*(5*I*B*a^2 + 2*I*(5*A + 7*C)*a*b + 7*I*B*b^2)*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt( 2)*(-5*I*B*a^2 - 2*I*(5*A + 7*C)*a*b - 7*I*B*b^2)*cos(d*x + c)^5*weierstra ssPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(I*(7*A + 9* C)*a^2 + 18*I*B*a*b + 3*I*(3*A + 5*C)*b^2)*cos(d*x + c)^5*weierstrassZeta( -4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqr t(2)*(-I*(7*A + 9*C)*a^2 - 18*I*B*a*b - 3*I*(3*A + 5*C)*b^2)*cos(d*x + c)^ 5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d *x + c))) - 2*(21*((7*A + 9*C)*a^2 + 18*B*a*b + 3*(3*A + 5*C)*b^2)*cos(d*x + c)^4 + 15*(5*B*a^2 + 2*(5*A + 7*C)*a*b + 7*B*b^2)*cos(d*x + c)^3 + 35*A *a^2 + 7*((7*A + 9*C)*a^2 + 18*B*a*b + 9*A*b^2)*cos(d*x + c)^2 + 45*(B*a^2 + 2*A*a*b)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) ^5)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)* *(11/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 1/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2/c os(d*x + c)^(11/2), x)
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 1/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2/c os(d*x + c)^(11/2), x)
Time = 3.05 (sec) , antiderivative size = 851, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2),x)
Output:
(2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((28*A*a^2*sin(c + d*x))/(c os(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (12*A*a^2*sin(c + d*x))/(c os(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2)) + (5*A*a^2*sin(c + d*x))/(co s(c + d*x)^(9/2)*(1 - cos(c + d*x)^2)^(1/2)) + (36*A*b^2*sin(c + d*x))/(co s(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (9*A*b^2*sin(c + d*x))/(cos (c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2)) + (36*C*a^2*sin(c + d*x))/(cos (c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (9*C*a^2*sin(c + d*x))/(cos( c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2)) + (45*C*b^2*sin(c + d*x))/(cos( c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (72*B*a*b*sin(c + d*x))/(cos( c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (18*B*a*b*sin(c + d*x))/(cos( c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2))))/(45*d) + (2*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((4*B*a^2*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (3*B*a^2*sin(c + d*x))/(cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (7*B*b^2*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (8*A*a*b*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - c os(c + d*x)^2)^(1/2)) + (6*A*a*b*sin(c + d*x))/(cos(c + d*x)^(7/2)*(1 - co s(c + d*x)^2)^(1/2)) + (14*C*a*b*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - co s(c + d*x)^2)^(1/2))))/(21*d) + (8*((B*a^2*sin(c + d*x))/(cos(c + d*x)^(3/ 2)*(1 - cos(c + d*x)^2)^(1/2)) + (2*A*a*b*sin(c + d*x))/(cos(c + d*x)^(3/2 )*(1 - cos(c + d*x)^2)^(1/2)))*hypergeom([-3/4, 1/2], 5/4, cos(c + d*x)...
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{2} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{2} c +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a \,b^{2}+2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a b c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) b^{3}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) b^{2} c \] Input:
int((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x )
Output:
int(sqrt(cos(c + d*x))/cos(c + d*x)**6,x)*a**3 + 3*int(sqrt(cos(c + d*x))/ cos(c + d*x)**5,x)*a**2*b + int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**2 *c + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a*b**2 + 2*int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a*b*c + int(sqrt(cos(c + d*x))/cos(c + d*x)**3 ,x)*b**3 + int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*b**2*c