\(\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [1029]
Optimal result
Integrand size = 43, antiderivative size = 629 \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}
\]
[Out]
2/45045*(520*B*a^3*b+4355*B*a*b^3-240*a^4*C+539*b^4*(13*A+11*C)-10*a^2*b^2*(143*A+124*C))*(a+b*cos(d*x+c))^(3/
2)*sin(d*x+c)/b^3/d+2/9009*(104*B*a^2*b+1053*B*b^3-48*a^3*C-2*a*b^2*(143*A+166*C))*(a+b*cos(d*x+c))^(5/2)*sin(
d*x+c)/b^3/d+2/1287*(143*A*b^2-52*B*a*b+24*C*a^2+121*C*b^2)*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b^3/d+2/143*(13*
B*b-6*C*a)*cos(d*x+c)*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b^2/d+2/13*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^(7/2)*sin(d
*x+c)/b/d+2/45045*(520*B*a^4*b+3705*B*a^2*b^3+8775*B*b^5-240*C*a^5-10*a^3*b^2*(143*A+94*C)+6*a*b^4*(2717*A+217
4*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^3/d+2/45045*(520*B*a^5*b+3315*B*a^3*b^3+48165*B*a*b^5-240*a^6*C+1617
*b^6*(13*A+11*C)-10*a^4*b^2*(143*A+76*C)+3*a^2*b^4*(13299*A+10223*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x
+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/b^4/d/((a+b*cos(d*x+c))/(
a+b))^(1/2)-2/45045*(a^2-b^2)*(520*B*a^4*b+3705*B*a^2*b^3+8775*B*b^5-240*C*a^5-10*a^3*b^2*(143*A+94*C)+6*a*b^4
*(2717*A+2174*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b
))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/b^4/d/(a+b*cos(d*x+c))^(1/2)
Rubi [A] (verified)
Time = 1.73 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.00, number of
steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {3128, 3102, 2832, 2831, 2742,
2740, 2734, 2732} \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{45045 b^3 d}-\frac {2 \left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}
\]
[In]
Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
[Out]
(2*(520*a^5*b*B + 3315*a^3*b^3*B + 48165*a*b^5*B - 240*a^6*C + 1617*b^6*(13*A + 11*C) - 10*a^4*b^2*(143*A + 76
*C) + 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(45045*b^
4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*
C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d
*x)/2, (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B
- 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4
5045*b^3*d) + (2*(520*a^3*b*B + 4355*a*b^3*B - 240*a^4*C + 539*b^4*(13*A + 11*C) - 10*a^2*b^2*(143*A + 124*C))
*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) + (2*(104*a^2*b*B + 1053*b^3*B - 48*a^3*C - 2*a*b^2*(1
43*A + 166*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(143*A*b^2 - 52*a*b*B + 24*a^2*C + 1
21*b^2*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) + (2*(13*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[
c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b
*d)
Rule 2732
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]
Rule 2734
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]
Rule 2740
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2742
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]
Rule 2831
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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]
Rule 2832
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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]
Rule 3102
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((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 + 1)/(b*f*(m + 2))), x] + Dist[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]
Rule 3128
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] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Rubi steps \begin{align*}
\text {integral}& = \frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {2 \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (2 a C+\frac {1}{2} b (13 A+11 C) \cos (c+d x)+\frac {1}{2} (13 b B-6 a C) \cos ^2(c+d x)\right ) \, dx}{13 b} \\ & = \frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {4 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{2} a (13 b B-6 a C)+\frac {1}{4} b (117 b B-10 a C) \cos (c+d x)+\frac {1}{4} \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{143 b^2} \\ & = \frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {8 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{8} b \left (1001 A b^2-130 a b B+60 a^2 C+847 b^2 C\right )+\frac {1}{8} \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) \cos (c+d x)\right ) \, dx}{1287 b^3} \\ & = \frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {16 \int (a+b \cos (c+d x))^{3/2} \left (-\frac {3}{16} b \left (130 a^2 b B-1755 b^3 B-60 a^3 C-a b^2 (1859 A+1423 C)\right )+\frac {1}{16} \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) \cos (c+d x)\right ) \, dx}{9009 b^3} \\ & = \frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {32 \int \sqrt {a+b \cos (c+d x)} \left (-\frac {3}{32} b \left (130 a^3 b B-13130 a b^3 B-60 a^4 C-539 b^4 (13 A+11 C)-5 a^2 b^2 (1573 A+1175 C)\right )+\frac {3}{32} \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \cos (c+d x)\right ) \, dx}{45045 b^3} \\ & = \frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {64 \int \frac {\frac {3}{64} b \left (130 a^4 b B+43095 a^2 b^3 B+8775 b^5 B-60 a^5 C+5 a^3 b^2 (4433 A+3337 C)+3 a b^4 (12441 A+10277 C)\right )+\frac {3}{64} \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{135135 b^3} \\ & = \frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}-\frac {\left (\left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{45045 b^4}+\frac {\left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{45045 b^4} \\ & = \frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d}+\frac {\left (\left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{45045 b^4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{45045 b^4 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\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 )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d} \\
\end{align*}
Mathematica [A] (verified)
Time = 5.49 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.80
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {32 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (130 a^4 b B+43095 a^2 b^3 B+8775 b^5 B-60 a^5 C+5 a^3 b^2 (4433 A+3337 C)+3 a b^4 (12441 A+10277 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-520 a^5 b B-3315 a^3 b^3 B-48165 a b^5 B+240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (4 \left (-2080 a^4 b B+121290 a^2 b^3 B+84825 b^5 B+960 a^5 C+10 a^3 b^2 (572 A+331 C)+3 a b^4 (71214 A+60793 C)\right ) \sin (c+d x)+b \left (\left (3120 a^3 b B+321880 a b^3 B-1440 a^4 C+120 a^2 b^2 (1430 A+1457 C)+77 b^4 (1976 A+1897 C)\right ) \sin (2 (c+d x))+5 b \left (2 \left (5876 a^2 b B+6669 b^3 B+60 a^3 C+a b^2 (10868 A+13939 C)\right ) \sin (3 (c+d x))+7 b \left (4 \left (143 A b^2+299 a b B+159 a^2 C+220 b^2 C\right ) \sin (4 (c+d x))+9 b ((26 b B+54 a C) \sin (5 (c+d x))+11 b C \sin (6 (c+d x)))\right )\right )\right )\right )}{720720 b^4 d \sqrt {a+b \cos (c+d x)}}
\]
[In]
Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
[Out]
(32*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(130*a^4*b*B + 43095*a^2*b^3*B + 8775*b^5*B - 60*a^5*C + 5*a^3*b^2
*(4433*A + 3337*C) + 3*a*b^4*(12441*A + 10277*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-520*a^5*b*B - 3315
*a^3*b^3*B - 48165*a*b^5*B + 240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299
*A + 10223*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*
(a + b*Cos[c + d*x])*(4*(-2080*a^4*b*B + 121290*a^2*b^3*B + 84825*b^5*B + 960*a^5*C + 10*a^3*b^2*(572*A + 331*
C) + 3*a*b^4*(71214*A + 60793*C))*Sin[c + d*x] + b*((3120*a^3*b*B + 321880*a*b^3*B - 1440*a^4*C + 120*a^2*b^2*
(1430*A + 1457*C) + 77*b^4*(1976*A + 1897*C))*Sin[2*(c + d*x)] + 5*b*(2*(5876*a^2*b*B + 6669*b^3*B + 60*a^3*C
+ a*b^2*(10868*A + 13939*C))*Sin[3*(c + d*x)] + 7*b*(4*(143*A*b^2 + 299*a*b*B + 159*a^2*C + 220*b^2*C)*Sin[4*(
c + d*x)] + 9*b*((26*b*B + 54*a*C)*Sin[5*(c + d*x)] + 11*b*C*Sin[6*(c + d*x)]))))))/(720720*b^4*d*Sqrt[a + b*C
os[c + d*x]])
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(3164\) vs. \(2(651)=1302\).
Time = 63.82 (sec) , antiderivative size = 3165, normalized size of antiderivative =
5.03
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| method | result | size |
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| default |
\(\text {Expression too large to display}\) |
\(3165\) |
| parts |
\(\text {Expression too large to display}\) |
\(3504\) |
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[In]
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)
[Out]
-2/45045*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1430*A*(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*b^2-1773
2*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x
+1/2*c),(-2*b/(a-b))^(1/2))*b^4-13984*a^3*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b
)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4+16302*a*A*b^6*(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))-760*C*(s
in(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*b^2-520*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/
2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6*b+21021*A*(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*b^6-39897*A*(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^5+1430*A*(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)*Elli
pticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b^2+48165*B*(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^5+520*B*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1
/2))*a^6*b-5070*B*a^2*b^5*(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)*Ell
ipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-3185*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*
c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^3+13044*C*a*b^6*(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))+39897*A*(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^4+1430*A*(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^3+17787*C*(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*b^6-
3315*B*(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^4+760*C*(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^3+30669*C*(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^4-30669
*C*(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^5+700*C*(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*b^2-48165*B*(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*b^6+3315*B*(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^3+240*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)
*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6*b-520*B*(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*b^2+(262080*B*b^7+766080
*C*a*b^6+1330560*C*b^7)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^7-465920*B*a*b^6-655200*B*b^7-45
0240*C*a^2*b^5-1915200*C*a*b^6-1798720*C*b^7)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(297440*A*a*b^6+320320*
A*b^7+284960*B*a^2*b^5+931840*B*a*b^6+739440*B*b^7+90240*C*a^3*b^4+900480*C*a^2*b^5+2159680*C*a*b^6+1379840*C*
b^7)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-194480*A*a^2*b^5-446160*A*a*b^6-296296*A*b^7-60320*B*a^3*b^4-42
7440*B*a^2*b^5-860080*B*a*b^6-453960*B*b^7+120*C*a^4*b^3-135360*C*a^3*b^4-828880*C*a^2*b^5-1324320*C*a*b^6-666
512*C*b^7)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(45760*A*a^3*b^4+194480*A*a^2*b^5+344344*A*a*b^6+136136*A*b
^7-260*B*a^4*b^3+60320*B*a^3*b^4+326560*B*a^2*b^5+394160*B*a*b^6+180180*B*b^7+120*C*a^5*b^2-120*C*a^4*b^3+1018
40*C*a^3*b^4+378640*C*a^2*b^5+522368*C*a*b^6+198352*C*b^7)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1430*A*a^
4*b^3-22880*A*a^3*b^4-95238*A*a^2*b^5-97812*A*a*b^6-24024*A*b^7+520*B*a^5*b^2+130*B*a^4*b^3-41730*B*a^3*b^4-92
040*B*a^2*b^5-86970*B*a*b^6-36270*B*b^7-240*C*a^6*b-60*C*a^5*b^2-760*C*a^4*b^3-28360*C*a^3*b^4-104466*C*a^2*b^
5-104304*C*a*b^6-27258*C*b^7)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-443520*C*b^7*cos(1/2*d*x+1/2*c)*sin(1/2*
d*x+1/2*c)^14-21021*A*(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
cE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^7-17787*C*(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))*b^7-240*C*(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^7+24
0*C*(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^7+8775*B*b^7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-
b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)))/b^4/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1
/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 929, normalized size of antiderivative = 1.48
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
[Out]
1/135135*(sqrt(2)*(-480*I*C*a^7 + 1040*I*B*a^6*b - 20*I*(143*A + 67*C)*a^5*b^2 + 6240*I*B*a^4*b^3 + 3*I*(4433*
A + 3761*C)*a^3*b^4 - 32955*I*B*a^2*b^5 - 3*I*(23309*A + 18973*C)*a*b^6 - 26325*I*B*b^7)*sqrt(b)*weierstrassPI
nverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)
/b) + sqrt(2)*(480*I*C*a^7 - 1040*I*B*a^6*b + 20*I*(143*A + 67*C)*a^5*b^2 - 6240*I*B*a^4*b^3 - 3*I*(4433*A + 3
761*C)*a^3*b^4 + 32955*I*B*a^2*b^5 + 3*I*(23309*A + 18973*C)*a*b^6 + 26325*I*B*b^7)*sqrt(b)*weierstrassPInvers
e(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(2)*(240*I*C*a^6*b - 520*I*B*a^5*b^2 + 10*I*(143*A + 76*C)*a^4*b^3 - 3315*I*B*a^3*b^4 - 3*I*(13299*A +
10223*C)*a^2*b^5 - 48165*I*B*a*b^6 - 1617*I*(13*A + 11*C)*b^7)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2
, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(
3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*sqrt(2)*(-240*I*C*a^6*b + 520*I*B*a^5*b^2 - 10*I*(143*A +
76*C)*a^4*b^3 + 3315*I*B*a^3*b^4 + 3*I*(13299*A + 10223*C)*a^2*b^5 + 48165*I*B*a*b^6 + 1617*I*(13*A + 11*C)*b
^7)*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)) + 6*(3465*
C*b^7*cos(d*x + c)^5 + 120*C*a^5*b^2 - 260*B*a^4*b^3 + 5*(143*A + 79*C)*a^3*b^4 + 13325*B*a^2*b^5 + (23309*A +
18973*C)*a*b^6 + 8775*B*b^7 + 315*(27*C*a*b^6 + 13*B*b^7)*cos(d*x + c)^4 + 35*(159*C*a^2*b^5 + 299*B*a*b^6 +
11*(13*A + 11*C)*b^7)*cos(d*x + c)^3 + 5*(15*C*a^3*b^4 + 1469*B*a^2*b^5 + (2717*A + 2209*C)*a*b^6 + 1053*B*b^7
)*cos(d*x + c)^2 - (90*C*a^4*b^3 - 195*B*a^3*b^4 - 15*(715*A + 543*C)*a^2*b^5 - 14885*B*a*b^6 - 539*(13*A + 11
*C)*b^7)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^5*d)
Sympy [F(-1)]
Timed out. \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out}
\]
[In]
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
[Out]
Timed out
Maxima [F]
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x }
\]
[In]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^2, x)
Giac [F]
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x }
\]
[In]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^2, x)
Mupad [F(-1)]
Timed out. \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x
\]
[In]
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
[Out]
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)