\(\int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \, dx\) [696]

Optimal result

Integrand size = 35, antiderivative size = 382 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d} \]

[Out]

2/195*(39*a^4*(5*A+3*C)+78*a^2*b^2*(9*A+7*C)+7*b^4*(13*A+11*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))/d+8/231*a*b*(11*a^2*(7*A+5*C)+5*b^2*(11*A+9*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))/d+2/6435*(192*a^4*C+77*b^4*(13*A+11*C)+11*a^2*
b^2*(637*A+491*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+4/9009*a*b*(1573*A*b^2+96*C*a^2+1259*C*b^2)*cos(d*x+c)^(5/2)*
sin(d*x+c)/d+2/1287*(48*a^2*C+11*b^2*(13*A+11*C))*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+16/143*a*C*
cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d+2/13*C*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^4*sin(d*x+c)/d+8/231
*a*b*(11*a^2*(7*A+5*C)+5*b^2*(11*A+9*C))*sin(d*x+c)*cos(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3129, 3128, 3112, 3102, 2827, 2719, 2715, 2720} \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6435 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d} \]

[In]

Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]

[Out]

(2*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2])/(195*d) + (8
*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a*b*(11*a^2*(7*A + 5*C)
+ 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^
2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (4*a*b*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)
*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(48*a^2*C + 11*b^2*(13*A + 11*C))*Cos[c + d*x]^(3/2)*(a + b*Co
s[c + d*x])^2*Sin[c + d*x])/(1287*d) + (16*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d)
+ (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

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]

Rule 2720

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]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!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])))

Rule 3129

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e +
f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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 ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2}{13} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+3 C)+\frac {1}{2} b (13 A+11 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {4}{143} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac {1}{4} a^2 (143 A+57 C)+\frac {1}{2} a b (143 A+113 C) \cos (c+d x)+\frac {1}{4} \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {8 \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {3}{8} a \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {1}{8} b \left (77 b^2 (13 A+11 C)+3 a^2 (1287 A+961 C)\right ) \cos (c+d x)+\frac {1}{4} a \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{1287} \\ & = \frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {16 \int \sqrt {\cos (c+d x)} \left (\frac {21}{16} a^2 \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {117}{4} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac {7}{16} \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^2(c+d x)\right ) \, dx}{9009} \\ & = \frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {32 \int \sqrt {\cos (c+d x)} \left (\frac {231}{32} \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right )+\frac {585}{8} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045} \\ & = \frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {1}{77} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {1}{231} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.83 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.74 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {7392 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+24960 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \sqrt {\cos (c+d x)} \left (154 \left (936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 b \left (312 a \left (44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right )+3744 a \left (11 A b^2+11 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+77 \left (52 A b^3+312 a^2 b C+89 b^3 C\right ) \cos (3 (c+d x))+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right ) \sin (c+d x)}{720720 d} \]

[In]

Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]

[Out]

(7392*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2] + 24960*a*
b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(936*a^4*C +
156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*b*(312*a*(44*a^2*(14*A + 13*C) + b^2*(572
*A + 531*C)) + 3744*a*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 77*(52*A*b^3 + 312*a^2*b*C + 89*b^3*
C)*Cos[3*(c + d*x)] + 6552*a*b^2*C*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*Sin[c + d*x])/(720720*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1016\) vs. \(2(406)=812\).

Time = 32.65 (sec) , antiderivative size = 1017, normalized size of antiderivative = 2.66

method	result	size
default	\(\text {Expression too large to display}\)	\(1017\)
parts	\(\text {Expression too large to display}\)	\(1446\)

[In]

int((a+cos(d*x+c)*b)^4*(A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^14*b^4+(1048320*C*a*b^3+1330560*C*b^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^4-960960*C*a^
2*b^2-2620800*C*a*b^3-1798720*C*b^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(411840*A*a*b^3+320320*A*b^4+411
840*C*a^3*b+1921920*C*a^2*b^2+2957760*C*a*b^3+1379840*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-432432*
A*a^2*b^2-617760*A*a*b^3-296296*A*b^4-72072*C*a^4-617760*C*a^3*b-1777776*C*a^2*b^2-1815840*C*a*b^3-666512*C*b^
4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(240240*A*a^3*b+432432*A*a^2*b^2+480480*A*a*b^3+136136*A*b^4+72072*
C*a^4+480480*C*a^3*b+816816*C*a^2*b^2+720720*C*a*b^3+198352*C*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1
20120*A*a^3*b-108108*A*a^2*b^2-137280*A*a*b^3-24024*A*b^4-18018*C*a^4-137280*C*a^3*b-144144*C*a^2*b^2-145080*C
*a*b^3-27258*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+60060*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*b+42900*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^3-45045*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si
n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^4-162162*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*b^2-21021*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^4+42900*C*a^3*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))+35100*C*a*b^3*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-27027*C*(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^4-126126*C*(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*b^2-17787*C*(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^4)/(-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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.08 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (3465 \, C b^{4} \cos \left (d x + c\right )^{5} + 16380 \, C a b^{3} \cos \left (d x + c\right )^{4} + 8580 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 3900 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 385 \, {\left (78 \, C a^{2} b^{2} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} + 2340 \, {\left (11 \, C a^{3} b + {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (117 \, C a^{4} + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 780 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 780 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{45045 \, d} \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/45045*(2*(3465*C*b^4*cos(d*x + c)^5 + 16380*C*a*b^3*cos(d*x + c)^4 + 8580*(7*A + 5*C)*a^3*b + 3900*(11*A + 9
*C)*a*b^3 + 385*(78*C*a^2*b^2 + (13*A + 11*C)*b^4)*cos(d*x + c)^3 + 2340*(11*C*a^3*b + (11*A + 9*C)*a*b^3)*cos
(d*x + c)^2 + 77*(117*C*a^4 + 78*(9*A + 7*C)*a^2*b^2 + 7*(13*A + 11*C)*b^4)*cos(d*x + c))*sqrt(cos(d*x + c))*s
in(d*x + c) - 780*sqrt(2)*(11*I*(7*A + 5*C)*a^3*b + 5*I*(11*A + 9*C)*a*b^3)*weierstrassPInverse(-4, 0, cos(d*x
+ c) + I*sin(d*x + c)) - 780*sqrt(2)*(-11*I*(7*A + 5*C)*a^3*b - 5*I*(11*A + 9*C)*a*b^3)*weierstrassPInverse(-
4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-39*I*(5*A + 3*C)*a^4 - 78*I*(9*A + 7*C)*a^2*b^2 - 7*I*(13
*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2
)*(39*I*(5*A + 3*C)*a^4 + 78*I*(9*A + 7*C)*a^2*b^2 + 7*I*(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrass
PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d

Sympy [F(-1)]

Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4*sqrt(cos(d*x + c)), x)

Giac [F]

\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4*sqrt(cos(d*x + c)), x)

Mupad [B] (verification not implemented)

Time = 3.42 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.77 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {136\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {23}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {11\,C\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {9\,C\,a^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {42\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{21945\,d}-\frac {2\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {165\,C\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {52\,C\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {36\,C\,a^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {77\,C\,b^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {630\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {168\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{1155\,d}-\frac {8\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {13\,C\,a^3\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {9\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a^3\,b\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{117\,d}+\frac {4\,A\,a^3\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,b^4\,{\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 {8\,A\,a\,b^3\,{\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 {160\,C\,a^3\,b\,{\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 {21}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{663\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {12\,A\,a^2\,b^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}} \]

[In]

int(cos(c + d*x)^(1/2)*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4,x)

[Out]

(2*A*a^4*ellipticE(c/2 + (d*x)/2, 2))/d - (136*hypergeom([1/2, 15/4], 23/4, cos(c + d*x)^2)*((11*C*a^4*cos(c +
d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (9*C*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2
)^(1/2) + (42*C*a^2*b^2*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2)))/(21945*d) - (2*hypergeom([1
/2, 15/4], 19/4, cos(c + d*x)^2)*((165*C*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (52*C*a
^4*cos(c + d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (36*C*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin
(c + d*x)^2)^(1/2) + (77*C*b^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (630*C*a^2*b^2*cos(c
+ d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (168*C*a^2*b^2*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c
+ d*x)^2)^(1/2)))/(1155*d) - (8*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2)*((13*C*a^3*b*cos(c + d*x)^(9/2)*s
in(c + d*x))/(sin(c + d*x)^2)^(1/2) + (9*C*a*b^3*cos(c + d*x)^(13/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (4
*C*a^3*b*cos(c + d*x)^(13/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2)))/(117*d) + (4*A*a^3*b*((2*cos(c + d*x)^(1/2
)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*b^4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeo
m([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (8*A*a*b^3*cos(c + d*x)^(9/2)*sin(c + d*
x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (160*C*a^3*b*cos(c + d*x)^(13/2
)*sin(c + d*x)*hypergeom([1/2, 13/4], 21/4, cos(c + d*x)^2))/(663*d*(sin(c + d*x)^2)^(1/2)) - (12*A*a^2*b^2*co
s(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))