\(\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [1030]
Optimal result
Integrand size = 41, antiderivative size = 510 \[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}
\]
[Out]
-2/3465*(110*B*a^2*b-539*B*b^3-40*a^3*C-5*a*b^2*(99*A+67*C))*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+2/693*(99
*A*b^2-22*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*a^4*C-75*b^4*(11*A+9*C)-15*a^2*b^2*(33*A+19*C))*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^5-15*a^3*b^2*(33*A+17*C)-15*a*b^4*(319*A+247*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^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*a^4*C-75*b^4*(11*A+9*C)-15*a^2*b^2*(
33*A+19*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^3/d/(a+b*cos(d*x+c))^(1/2)
Rubi [A] (verified)
Time = 1.24 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of
steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3128, 3102, 2832, 2831, 2742,
2740, 2734, 2732} \[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \left (8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \sin (c+d x) \left (-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 \sin (c+d x) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3465 b^2 d}+\frac {2 \left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}
\]
[In]
Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + 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 - 15*a^3*b^2*(33*A + 17*C) - 15*a*b^4*(319*A + 247*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 - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))
*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(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 - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Co
s[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 - 5*a*b^2*(99*A + 67*C))*(a +
b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) + (2*(99*A*b^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 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 (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 {1}{2} b (11 A+9 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 (99 A b^2-22 a b B+8 a^2 C+81 b^2 C\right ) \cos (c+d x)\right ) \, dx}{99 b^2} \\ & = \frac {2 \left (99 A b^2-22 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 (165 A b^2+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-5 a b^2 (99 A+67 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-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 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+10 a b^2 (132 A+101 C)\right )-\frac {3}{16} \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 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+75 b^4 (11 A+9 C)+15 a^2 b^2 (297 A+221 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-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 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-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-15 a^3 b^2 (33 A+17 C)-15 a b^4 (319 A+247 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-75 b^4 (11 A+9 C)-15 a^2 b^2 (33 A+19 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-5 a b^2 (99 A+67 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2-22 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 = 3.97 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.79
\[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+75 b^4 (11 A+9 C)+15 a^2 b^2 (297 A+221 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-110 a^4 b B+3069 a^2 b^3 B+1617 b^5 B+40 a^5 C+15 a^3 b^2 (33 A+17 C)+15 a b^4 (319 A+247 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (880 a^3 b B+32868 a b^3 B-320 a^4 C+60 a^2 b^2 (396 A+311 C)+30 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 b B+1463 b^3 B+30 a^3 C+5 a b^2 (594 A+619 C)\right ) \sin (2 (c+d x))+5 b \left (\left (396 A b^2+836 a b B+452 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((22 b B+46 a C) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )}{27720 b^3 d \sqrt {a+b \cos (c+d x)}}
\]
[In]
Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + 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 + 75*b^4*(11*A + 9*C) + 15
*a^2*b^2*(297*A + 221*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 + 15*a^3*b^2*(33*A + 17*C) + 15*a*b^4*(319*A + 247*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 + 60*a^2*b^2*(396*A + 311*C) + 30*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*(1650*a^2*b*B + 1463*b^3*B + 30
*a^3*C + 5*a*b^2*(594*A + 619*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 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)])))))/(27720*b^3*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. \(2602\) vs. \(2(536)=1072\).
Time = 35.11 (sec) , antiderivative size = 2603, normalized size of antiderivative =
5.10
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2603\) |
| parts |
\(\text {Expression too large to display}\) |
\(2964\) |
| | |
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[In]
int(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)
[Out]
-2/3465*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((-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)+(7920*A*b^6+22880*B*a*b^5+24640*B*b^6+21920*C*a^2*b^4+71680*C*a*b^5+5
6880*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-15840*A*a*b^5-11880*A*b^6-14960*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)+
(11880*A*a^2*b^4+15840*A*a*b^5+9240*A*b^6+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*cos(1/2*d*x+1/2*c)+(-2970*A*a
^3*b^3-5940*A*a^2*b^4-5610*A*a*b^5-2640*A*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^4-6690*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)+675*C*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)*Ellipt
icF(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)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6-1617*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+825*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))+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)*E
llipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6+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))-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^4-245*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)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+4785*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
^4+20160*C*b^6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-4785*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^5+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^3-495*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)*EllipticF
(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2-3705*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^5-3069*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^5-110*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-1364*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^3*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)*EllipticE(cos(1/2*d*x+1/2*c
),(-2*b/(a-b))^(1/2))*a^4*b^2+495*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^2-495*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^3-390*a^2*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)*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)*El
lipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b-330*A*a^2*(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+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+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(co
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b)/b^3/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/s
in(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 = 0.20 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.59
\[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
[Out]
1/10395*(sqrt(2)*(80*I*C*a^6 - 220*I*B*a^5*b + 30*I*(33*A + 16*C)*a^4*b^2 + 1023*I*B*a^3*b^3 - 15*I*(253*A + 1
69*C)*a^2*b^4 - 5379*I*B*a*b^5 - 225*I*(11*A + 9*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 - 30*I*(33*A + 16*C)*a^4*b^2 - 1023*I*B*a^3*b^3 + 15*I*(253*A + 169*C)*a^2*b^4 + 5379*I*B*a*b^5 + 2
25*I*(11*A + 9*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 - 15*I*(33*A + 17
*C)*a^3*b^3 - 3069*I*B*a^2*b^4 - 15*I*(319*A + 247*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)) - 3*sqrt(2)*(40*I*C*a^5*b - 110*I*B*a^4*b^2 + 15
*I*(33*A + 17*C)*a^3*b^3 + 3069*I*B*a^2*b^4 + 15*I*(319*A + 247*C)*a*b^5 + 1617*I*B*b^6)*sqrt(b)*weierstrassZe
ta(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 + 5*(297*A + 205*C)*a^2*b^4 + 1793*B*a*b^5 + 75*(11*A + 9*C)*b^6 + 35*(23*C*a*b^5 + 1
1*B*b^6)*cos(d*x + c)^3 + 5*(113*C*a^2*b^4 + 209*B*a*b^5 + 9*(11*A + 9*C)*b^6)*cos(d*x + c)^2 + (15*C*a^3*b^3
+ 825*B*a^2*b^4 + 5*(297*A + 229*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. \[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out}
\]
[In]
integrate(cos(d*x+c)*(a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
[Out]
Timed out
Maxima [F]
\[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x }
\]
[In]
integrate(cos(d*x+c)*(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), x)
Giac [F]
\[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x }
\]
[In]
integrate(cos(d*x+c)*(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), x)
Mupad [F(-1)]
Timed out. \[
\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x
\]
[In]
int(cos(c + d*x)*(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)*(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)