Integrand size = 33, antiderivative size = 264 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {10 \left (9 b^2 B+11 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (9 a^2 A+7 A b^2+14 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 A b+13 a B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b B \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d} \] Output:
2/15*(9*A*a^2+7*A*b^2+14*B*a*b)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+10 /231*(9*b^2*B+11*a*(2*A*b+B*a))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+1 0/231*(9*b^2*B+11*a*(2*A*b+B*a))*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/45*(9*A*a ^2+7*A*b^2+14*B*a*b)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/77*(9*b^2*B+11*a*(2*A *b+B*a))*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/99*b*(11*A*b+13*B*a)*cos(d*x+c)^( 7/2)*sin(d*x+c)/d+2/11*b*B*cos(d*x+c)^(7/2)*(a+b*cos(d*x+c))*sin(d*x+c)/d
Time = 3.35 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.74 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {3696 \left (9 a^2 A+7 A b^2+14 a b B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+1200 \left (22 a A b+11 a^2 B+9 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \sqrt {\cos (c+d x)} \left (154 \left (36 a^2 A+43 A b^2+86 a b B\right ) \cos (c+d x)+180 \left (22 a A b+11 a^2 B+16 b^2 B\right ) \cos (2 (c+d x))+770 b (A b+2 a B) \cos (3 (c+d x))+15 \left (1144 a A b+572 a^2 B+531 b^2 B+21 b^2 B \cos (4 (c+d x))\right )\right ) \sin (c+d x)}{27720 d} \] Input:
Integrate[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x ]
Output:
(3696*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(c + d*x)/2, 2] + 1200*(22* a*A*b + 11*a^2*B + 9*b^2*B)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x ]]*(154*(36*a^2*A + 43*A*b^2 + 86*a*b*B)*Cos[c + d*x] + 180*(22*a*A*b + 11 *a^2*B + 16*b^2*B)*Cos[2*(c + d*x)] + 770*b*(A*b + 2*a*B)*Cos[3*(c + d*x)] + 15*(1144*a*A*b + 572*a^2*B + 531*b^2*B + 21*b^2*B*Cos[4*(c + d*x)]))*Si n[c + d*x])/(27720*d)
Time = 1.11 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.88, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3469, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 3042, 3115, 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.
| \(\displaystyle \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3469 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \cos ^{\frac {5}{2}}(c+d x) \left (b (11 A b+13 a B) \cos ^2(c+d x)+\left (9 B b^2+11 a (2 A b+a B)\right ) \cos (c+d x)+a (11 a A+7 b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \cos ^{\frac {5}{2}}(c+d x) \left (b (11 A b+13 a B) \cos ^2(c+d x)+\left (9 B b^2+11 a (2 A b+a B)\right ) \cos (c+d x)+a (11 a A+7 b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (b (11 A b+13 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (9 B b^2+11 a (2 A b+a B)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 a A+7 b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {5}{2}}(c+d x) \left (11 \left (9 A a^2+14 b B a+7 A b^2\right )+9 \left (9 B b^2+11 a (2 A b+a B)\right ) \cos (c+d x)\right )dx+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \cos ^{\frac {5}{2}}(c+d x) \left (11 \left (9 A a^2+14 b B a+7 A b^2\right )+9 \left (9 B b^2+11 a (2 A b+a B)\right ) \cos (c+d x)\right )dx+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (11 \left (9 A a^2+14 b B a+7 A b^2\right )+9 \left (9 B b^2+11 a (2 A b+a B)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \int \cos ^{\frac {5}{2}}(c+d x)dx+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \int \cos ^{\frac {7}{2}}(c+d x)dx\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}dx\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \left (\frac {5}{7} \int \cos ^{\frac {3}{2}}(c+d x)dx+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \left (\frac {5}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 \left (9 a^2 A+14 a b B+7 A b^2\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 \left (11 a (a B+2 A b)+9 b^2 B\right ) \left (\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {5}{7} \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )+\frac {2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}\) |
Input:
Int[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]
Output:
(2*b*B*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d) + ((2* b*(11*A*b + 13*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (11*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*((6*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d* x]^(3/2)*Sin[c + d*x])/(5*d)) + 9*(9*b^2*B + 11*a*(2*A*b + a*B))*((2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (5*((2*EllipticF[(c + d*x)/2, 2])/(3*d ) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/7))/9)/11
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[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-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 - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*( m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin [e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !(IGt Q[n, 1] && ( !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. \(665\) vs. \(2(243)=486\).
Time = 57.68 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.52
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (20160 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} b^{2}+\left (-12320 A \,b^{2}-24640 B a b -50400 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (15840 A a b +24640 A \,b^{2}+7920 a^{2} B +49280 B a b +56880 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-5544 a^{2} A -23760 A a b -22792 A \,b^{2}-11880 a^{2} B -45584 B a b -34920 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5544 a^{2} A +18480 A a b +10472 A \,b^{2}+9240 a^{2} B +20944 B a b +13860 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1386 a^{2} A -5280 A a b -1848 A \,b^{2}-2640 a^{2} B -3696 B a b -2790 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1650 A a b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2079 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-1617 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+825 a^{2} B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+675 B \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3234 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b \right )}{3465 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(666\) |
| parts | \(\text {Expression too large to display}\) | \(866\) |
Input:
int(cos(d*x+c)^(5/2)*(a+cos(d*x+c)*b)^2*(A+B*cos(d*x+c)),x,method=_RETURNV ERBOSE)
Output:
-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*B*c os(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12*b^2+(-12320*A*b^2-24640*B*a*b-5040 0*B*b^2)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(15840*A*a*b+24640*A*b^2 +7920*B*a^2+49280*B*a*b+56880*B*b^2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2* c)+(-5544*A*a^2-23760*A*a*b-22792*A*b^2-11880*B*a^2-45584*B*a*b-34920*B*b^ 2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(5544*A*a^2+18480*A*a*b+10472*A *b^2+9240*B*a^2+20944*B*a*b+13860*B*b^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+ 1/2*c)+(-1386*A*a^2-5280*A*a*b-1848*A*b^2-2640*B*a^2-3696*B*a*b-2790*B*b^2 )*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1650*A*a*b*(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1 /2))-2079*A*(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))*a^2-1617*A*(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)) *b^2+825*a^2*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/ 2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+675*B*b^2*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2 ))-3234*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*El lipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2* d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.13 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \, {\left (315 \, B b^{2} \cos \left (d x + c\right )^{4} + 385 \, {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} + 825 \, B a^{2} + 1650 \, A a b + 675 \, B b^{2} + 45 \, {\left (11 \, B a^{2} + 22 \, A a b + 9 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, A a^{2} + 14 \, B a b + 7 \, A b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 75 \, \sqrt {2} {\left (11 i \, B a^{2} + 22 i \, A a b + 9 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 75 \, \sqrt {2} {\left (-11 i \, B a^{2} - 22 i \, A a b - 9 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-9 i \, A a^{2} - 14 i \, B a b - 7 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (9 i \, A a^{2} + 14 i \, B a b + 7 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \] Input:
integrate(cos(d*x+c)^(5/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorith m="fricas")
Output:
1/3465*(2*(315*B*b^2*cos(d*x + c)^4 + 385*(2*B*a*b + A*b^2)*cos(d*x + c)^3 + 825*B*a^2 + 1650*A*a*b + 675*B*b^2 + 45*(11*B*a^2 + 22*A*a*b + 9*B*b^2) *cos(d*x + c)^2 + 77*(9*A*a^2 + 14*B*a*b + 7*A*b^2)*cos(d*x + c))*sqrt(cos (d*x + c))*sin(d*x + c) - 75*sqrt(2)*(11*I*B*a^2 + 22*I*A*a*b + 9*I*B*b^2) *weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 75*sqrt(2)*(- 11*I*B*a^2 - 22*I*A*a*b - 9*I*B*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-9*I*A*a^2 - 14*I*B*a*b - 7*I*A*b^2)*w eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2)*(9*I*A*a^2 + 14*I*B*a*b + 7*I*A*b^2)*weierstrassZeta( -4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(5/2)*(a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)),x)
Output:
Timed out
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(5/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(5/2), x)
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(5/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(5/2), x)
Time = 24.77 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.04 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a\,b\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2,x)
Output:
- (2*A*a^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos (c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^2*cos(c + d*x)^(9/2)*s in(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x )^2)^(1/2)) - (2*A*b^2*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11 /4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*B*b^2*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/ (13*d*(sin(c + d*x)^2)^(1/2)) - (4*A*a*b*cos(c + d*x)^(9/2)*sin(c + d*x)*h ypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (4*B*a*b*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, co s(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2))
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) b^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{3} \] Input:
int(cos(d*x+c)^(5/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x)
Output:
int(sqrt(cos(c + d*x))*cos(c + d*x)**5,x)*b**3 + 3*int(sqrt(cos(c + d*x))* cos(c + d*x)**4,x)*a*b**2 + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a* *2*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**3