\(\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [827]

Optimal result

Integrand size = 40, antiderivative size = 462 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \]

[Out]

-2/3465*(110*B*a^2*b-539*B*b^3-40*C*a^3-335*C*a*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d-2/693*(22*B*a*b-8
*C*a^2-81*C*b^2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^2/d+2/99*(11*B*b-4*C*a)*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)
/b^2/d+2/11*C*cos(d*x+c)*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b/d-2/3465*(110*B*a^3*b-1254*B*a*b^3-40*C*a^4-285*C
*a^2*b^2-675*C*b^4)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d-2/3465*(110*B*a^4*b-3069*B*a^2*b^3-1617*B*b^5-40*C
*a^5-255*C*a^3*b^2-3705*C*a*b^4)*(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^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/3465*(a^2-b^2)*(110*B*a
^3*b-1254*B*a*b^3-40*C*a^4-285*C*a^2*b^2-675*C*b^4)*(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^3/d/(a+b*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {3108, 3069, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (-8 a^2 C+22 a b B-81 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \left (-40 a^3 C+110 a^2 b B-335 a b^2 C-539 b^3 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 \left (-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 b^2 d}+\frac {2 \left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-40 a^5 C+110 a^4 b B-255 a^3 b^2 C-3069 a^2 b^3 B-3705 a b^4 C-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]

[In]

Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(-2*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 255*a^3*b^2*C - 3705*a*b^4*C)*Sqrt[a + b*Cos[c + d
*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(
110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Elliptic
F[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4
*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^
3*B - 40*a^3*C - 335*a*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*b*B - 8*a^2*C -
81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(7/
2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*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 3069

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-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] + Dist[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] &&  !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

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 3108

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] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])
^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x)) \, dx \\ & = \frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {2 \int (a+b \cos (c+d x))^{5/2} \left (a C+\frac {9}{2} b C \cos (c+d x)+\frac {1}{2} (11 b B-4 a C) \cos ^2(c+d x)\right ) \, dx}{11 b} \\ & = \frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {4 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{4} b (77 b B-10 a C)-\frac {1}{4} \left (22 a b B-8 a^2 C-81 b^2 C\right ) \cos (c+d x)\right ) \, dx}{99 b^2} \\ & = -\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {8 \int (a+b \cos (c+d x))^{3/2} \left (\frac {3}{8} b \left (143 a b B-10 a^2 C+135 b^2 C\right )-\frac {1}{8} \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) \cos (c+d x)\right ) \, dx}{693 b^2} \\ & = -\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {16 \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{16} b \left (605 a^2 b B+539 b^3 B-10 a^3 C+1010 a b^2 C\right )-\frac {3}{16} \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \cos (c+d x)\right ) \, dx}{3465 b^2} \\ & = -\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {32 \int \frac {\frac {3}{32} b \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+3315 a^2 b^2 C+675 b^4 C\right )-\frac {3}{32} \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{10395 b^2} \\ & = -\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {\left (\left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3465 b^3}-\frac {\left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3465 b^3} \\ & = -\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}-\frac {\left (\left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 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}{3465 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 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}{3465 b^3 \sqrt {a+b \cos (c+d x)}} \\ & = -\frac {2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 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 )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.52 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.77 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+3315 a^2 b^2 C+675 b^4 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-110 a^4 b B+3069 a^2 b^3 B+1617 b^5 B+40 a^5 C+255 a^3 b^2 C+3705 a b^4 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (880 a^3 b B+32868 a b^3 B-320 a^4 C+18660 a^2 b^2 C+13050 b^4 C\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 b B+1463 b^3 B+30 a^3 C+3095 a b^2 C\right ) \sin (2 (c+d x))+5 b \left (\left (836 a b B+452 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((22 b B+46 a C) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )}{27720 b^3 d \sqrt {a+b \cos (c+d x)}} \]

[In]

Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(1705*a^3*b*B + 2871*a*b^3*B + 10*a^4*C + 3315*a^2*b^2*C + 675*b^4
*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b
^2*C + 3705*a*b^4*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])*((880*a^3*b*B + 32868*a*b^3*B - 320*a^4*C + 18660*a^2*b^2*C + 13050*b^4*C)*Sin[c +
d*x] + b*(4*(1650*a^2*b*B + 1463*b^3*B + 30*a^3*C + 3095*a*b^2*C)*Sin[2*(c + d*x)] + 5*b*((836*a*b*B + 452*a^2
*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*((22*b*B + 46*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(2772
0*b^3*d*Sqrt[a + b*Cos[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1982\) vs. \(2(488)=976\).

Time = 29.41 (sec) , antiderivative size = 1983, normalized size of antiderivative = 4.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1983\)
parts	\(\text {Expression too large to display}\)	\(2137\)

[In]

int(cos(d*x+c)*(a+cos(d*x+c)*b)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(675*b^6*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))+40*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^6-245*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)*Ellip
ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2-390*a^2*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^4-40*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^5*b+255*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^2-255*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^3+3705*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^4-
3705*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^5-40*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^6+20160*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12
*b^6+3069*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^3-3069*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^4+1617*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^5+11
0*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^2-110*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-1617*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))*b^6+(-110*B*a^4*b^2
-1760*B*a^3*b^3-7326*B*a^2*b^4-7524*B*a*b^5-1848*B*b^6+40*C*a^5*b+10*C*a^4*b^2-3210*C*a^3*b^3-7080*C*a^2*b^4-6
690*C*a*b^5-2790*C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(-12320*B*b^6-35840*C*a*b^5-50400*C*b^6)*sin(1
/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(22880*B*a*b^5+24640*B*b^6+21920*C*a^2*b^4+71680*C*a*b^5+56880*C*b^6)*sin(
1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-14960*B*a^2*b^4-34320*B*a*b^5-22792*B*b^6-4640*C*a^3*b^3-32880*C*a^2*b^4
-66160*C*a*b^5-34920*C*b^6)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(3520*B*a^3*b^3+14960*B*a^2*b^4+26488*B*a*
b^5+10472*B*b^6-20*C*a^4*b^2+4640*C*a^3*b^3+25120*C*a^2*b^4+30320*C*a*b^5+13860*C*b^6)*sin(1/2*d*x+1/2*c)^4*co
s(1/2*d*x+1/2*c)+1254*B*a*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)
*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1364*B*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^3+110*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^5*b)/b^3/(-2*sin(1/2*d*x+1/2*c)^4*b+(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 = 0.17 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.57 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (80 i \, C a^{6} - 220 i \, B a^{5} b + 480 i \, C a^{4} b^{2} + 1023 i \, B a^{3} b^{3} - 2535 i \, C a^{2} b^{4} - 5379 i \, B a b^{5} - 2025 i \, C b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-80 i \, C a^{6} + 220 i \, B a^{5} b - 480 i \, C a^{4} b^{2} - 1023 i \, B a^{3} b^{3} + 2535 i \, C a^{2} b^{4} + 5379 i \, B a b^{5} + 2025 i \, C b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-40 i \, C a^{5} b + 110 i \, B a^{4} b^{2} - 255 i \, C a^{3} b^{3} - 3069 i \, B a^{2} b^{4} - 3705 i \, C a b^{5} - 1617 i \, B b^{6}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (40 i \, C a^{5} b - 110 i \, B a^{4} b^{2} + 255 i \, C a^{3} b^{3} + 3069 i \, B a^{2} b^{4} + 3705 i \, C a b^{5} + 1617 i \, B b^{6}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (315 \, C b^{6} \cos \left (d x + c\right )^{4} - 20 \, C a^{4} b^{2} + 55 \, B a^{3} b^{3} + 1025 \, C a^{2} b^{4} + 1793 \, B a b^{5} + 675 \, C b^{6} + 35 \, {\left (23 \, C a b^{5} + 11 \, B b^{6}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (113 \, C a^{2} b^{4} + 209 \, B a b^{5} + 81 \, C b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, C a^{3} b^{3} + 825 \, B a^{2} b^{4} + 1145 \, C a b^{5} + 539 \, B b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{10395 \, b^{4} d} \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/10395*(sqrt(2)*(80*I*C*a^6 - 220*I*B*a^5*b + 480*I*C*a^4*b^2 + 1023*I*B*a^3*b^3 - 2535*I*C*a^2*b^4 - 5379*I*
B*a*b^5 - 2025*I*C*b^6)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*
(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(2)*(-80*I*C*a^6 + 220*I*B*a^5*b - 480*I*C*a^4*b^2 - 10
23*I*B*a^3*b^3 + 2535*I*C*a^2*b^4 + 5379*I*B*a*b^5 + 2025*I*C*b^6)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*
b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3*sqrt(2)*(-40*I
*C*a^5*b + 110*I*B*a^4*b^2 - 255*I*C*a^3*b^3 - 3069*I*B*a^2*b^4 - 3705*I*C*a*b^5 - 1617*I*B*b^6)*sqrt(b)*weier
strassZeta(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)*(40*I*C*a^5*b -
110*I*B*a^4*b^2 + 255*I*C*a^3*b^3 + 3069*I*B*a^2*b^4 + 3705*I*C*a*b^5 + 1617*I*B*b^6)*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*(315*C*b^6*cos(d*x + c)^4 - 20*C*
a^4*b^2 + 55*B*a^3*b^3 + 1025*C*a^2*b^4 + 1793*B*a*b^5 + 675*C*b^6 + 35*(23*C*a*b^5 + 11*B*b^6)*cos(d*x + c)^3
+ 5*(113*C*a^2*b^4 + 209*B*a*b^5 + 81*C*b^6)*cos(d*x + c)^2 + (15*C*a^3*b^3 + 825*B*a^2*b^4 + 1145*C*a*b^5 +
539*B*b^6)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^4*d)

Sympy [F(-1)]

Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))**(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(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))*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c), x)

Giac [F]

\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(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))*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c), x)

Mupad [F(-1)]

Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

int(cos(c + d*x)*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2),x)

[Out]

int(cos(c + d*x)*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x)