\(\int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \, dx\) [689]

Optimal result

Integrand size = 35, antiderivative size = 295 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{165 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d} \]

[Out]

2/5*a*(a^2*(5*A+3*C)+b^2*(9*A+7*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+2/231*b*(33*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)*Elli
pticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/165*a*(99*A*b^2+8*C*a^2+77*C*b^2)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/231*b*
(8*a^2*C+3*b^2*(11*A+9*C))*cos(d*x+c)^(5/2)*sin(d*x+c)/d+4/33*a*C*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*sin(d*x+
c)/d+2/11*C*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d+2/231*b*(33*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 = 0.88 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{231 d}+\frac {2 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{165 d}+\frac {2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{33 d} \]

[In]

Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]

[Out]

(2*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(
11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*b*(33*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*a*(99*A*b^2 + 8*a^2*C + 77*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(165*d) +
(2*b*(8*a^2*C + 3*b^2*(11*A + 9*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(231*d) + (4*a*C*Cos[c + d*x]^(3/2)*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(33*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*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))^3 \sin (c+d x)}{11 d}+\frac {2}{11} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac {1}{2} a (11 A+3 C)+\frac {1}{2} b (11 A+9 C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4}{99} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {9}{4} a^2 (11 A+5 C)+\frac {3}{2} a b (33 A+25 C) \cos (c+d x)+\frac {3}{4} \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {8}{693} \int \sqrt {\cos (c+d x)} \left (\frac {63}{8} a^3 (11 A+5 C)+\frac {9}{8} b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac {21}{8} a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{165 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {16 \int \sqrt {\cos (c+d x)} \left (\frac {693}{16} a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right )+\frac {45}{16} b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465} \\ & = \frac {2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{165 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {1}{5} \left (a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} \left (b \left (33 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 {2 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (33 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 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{165 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {1}{231} \left (b \left (33 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 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b \left (33 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 b \left (33 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 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{165 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.73 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3696 \left (a^3 (5 A+3 C)+a b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+80 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \sqrt {\cos (c+d x)} \left (154 a \left (36 A b^2+12 a^2 C+43 b^2 C\right ) \cos (c+d x)+5 b \left (1848 a^2 A+572 A b^2+1716 a^2 C+531 b^2 C+12 \left (11 A b^2+33 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 a b C \cos (3 (c+d x))+21 b^2 C \cos (4 (c+d x))\right )\right ) \sin (c+d x)}{9240 d} \]

[In]

Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]

[Out]

(3696*(a^3*(5*A + 3*C) + a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 80*b*(33*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*a*(36*A*b^2 + 12*a^2*C + 43*b^2*C)*Cos[c + d*x]
+ 5*b*(1848*a^2*A + 572*A*b^2 + 1716*a^2*C + 531*b^2*C + 12*(11*A*b^2 + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)]
+ 154*a*b*C*Cos[3*(c + d*x)] + 21*b^2*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(9240*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(792\) vs. \(2(323)=646\).

Time = 28.70 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.69

method	result	size
default	\(\text {Expression too large to display}\)	\(793\)
parts	\(\text {Expression too large to display}\)	\(1190\)

[In]

int((a+cos(d*x+c)*b)^3*(A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/1155*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(6720*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^
12*b^3+(-12320*C*a*b^2-16800*C*b^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(2640*A*b^3+7920*C*a^2*b+24640*C*
a*b^2+18960*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-5544*A*a*b^2-3960*A*b^3-1848*C*a^3-11880*C*a^2*b-
22792*C*a*b^2-11640*C*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(4620*A*a^2*b+5544*A*a*b^2+3080*A*b^3+1848*
C*a^3+9240*C*a^2*b+10472*C*a*b^2+4620*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2310*A*a^2*b-1386*A*a*b
^2-880*A*b^3-462*C*a^3-2640*C*a^2*b-1848*C*a*b^2-930*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1155*A*a^2
*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+275*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))-1155*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^3-2079*A
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+825
*C*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2
25*C*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))-6
93*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^3-1
617*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*b^
2)/(-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 = 0.15 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.14 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (105 \, C b^{3} \cos \left (d x + c\right )^{4} + 385 \, C a b^{2} \cos \left (d x + c\right )^{3} + 165 \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 25 \, {\left (11 \, A + 9 \, C\right )} b^{3} + 15 \, {\left (33 \, C a^{2} b + {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (3 \, C a^{3} + {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, \sqrt {2} {\left (33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 3 \, C\right )} a^{3} - i \, {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (i \, {\left (5 \, A + 3 \, C\right )} a^{3} + i \, {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{1155 \, d} \]

[In]

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/1155*(2*(105*C*b^3*cos(d*x + c)^4 + 385*C*a*b^2*cos(d*x + c)^3 + 165*(7*A + 5*C)*a^2*b + 25*(11*A + 9*C)*b^3
+ 15*(33*C*a^2*b + (11*A + 9*C)*b^3)*cos(d*x + c)^2 + 77*(3*C*a^3 + (9*A + 7*C)*a*b^2)*cos(d*x + c))*sqrt(cos
(d*x + c))*sin(d*x + c) - 5*sqrt(2)*(33*I*(7*A + 5*C)*a^2*b + 5*I*(11*A + 9*C)*b^3)*weierstrassPInverse(-4, 0,
cos(d*x + c) + I*sin(d*x + c)) - 5*sqrt(2)*(-33*I*(7*A + 5*C)*a^2*b - 5*I*(11*A + 9*C)*b^3)*weierstrassPInver
se(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-I*(5*A + 3*C)*a^3 - I*(9*A + 7*C)*a*b^2)*weierstrassZ
eta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2)*(I*(5*A + 3*C)*a^3 + I*(9*
A + 7*C)*a*b^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d

Sympy [F(-1)]

Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^3*(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)^3*sqrt(cos(d*x + c)), x)

Giac [F]

\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^3*(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)^3*sqrt(cos(d*x + c)), x)

Mupad [B] (verification not implemented)

Time = 2.85 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.14 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2\,\left (A\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^2\,b\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {2\,A\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^2\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

[In]

int(cos(c + d*x)^(1/2)*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3,x)

[Out]

(2*(A*a^3*ellipticE(c/2 + (d*x)/2, 2) + A*a^2*b*ellipticF(c/2 + (d*x)/2, 2) + A*a^2*b*cos(c + d*x)^(1/2)*sin(c
+ d*x)))/d - (2*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)) - (2*C*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*
d*(sin(c + d*x)^2)^(1/2)) - (2*C*b^3*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x
)^2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (6*A*a*b^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, c
os(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^2*b*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4]
, 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (6*C*a*b^2*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom(
[1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2))