\(\int \frac {\cos (c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx\) [1041]

Optimal result

Integrand size = 41, antiderivative size = 258 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \left (15 A b^2-10 a b B+8 a^2 C+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (10 a^2 b B+5 b^3 B-8 a^3 C-a b^2 (15 A+7 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 (5 b B-4 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac {2 C \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b d} \] Output:

2/15*(15*A*b^2-10*B*a*b+8*C*a^2+9*C*b^2)*(a+b*cos(d*x+c))^(1/2)*EllipticE( 
sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^3/d/((a+b*cos(d*x+c))/(a+b)) 
^(1/2)+2/15*(10*B*a^2*b+5*B*b^3-8*a^3*C-a*b^2*(15*A+7*C))*((a+b*cos(d*x+c) 
)/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b^3/ 
d/(a+b*cos(d*x+c))^(1/2)+2/15*(5*B*b-4*C*a)*(a+b*cos(d*x+c))^(1/2)*sin(d*x 
+c)/b^2/d+2/5*C*cos(d*x+c)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.72 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 (5 b B+2 a C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (15 A b^2-10 a b B+8 a^2 C+9 b^2 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+2 b (a+b \cos (c+d x)) (5 b B-4 a C+3 b C \cos (c+d x)) \sin (c+d x)}{15 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:

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

Output:

(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(5*b*B + 2*a*C)*EllipticF[(c + 
d*x)/2, (2*b)/(a + b)] + (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*((a + b 
)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/( 
a + b)])) + 2*b*(a + b*Cos[c + d*x])*(5*b*B - 4*a*C + 3*b*C*Cos[c + d*x])* 
Sin[c + d*x])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 3528, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {2 \int \frac {(5 b B-4 a C) \cos ^2(c+d x)+b (5 A+3 C) \cos (c+d x)+2 a C}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(5 b B-4 a C) \cos ^2(c+d x)+b (5 A+3 C) \cos (c+d x)+2 a C}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(5 b B-4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (5 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \int \frac {b (5 b B+2 a C)+\left (8 C a^2-10 b B a+15 A b^2+9 b^2 C\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {b (5 b B+2 a C)+\left (8 C a^2-10 b B a+15 A b^2+9 b^2 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {b (5 b B+2 a C)+\left (8 C a^2-10 b B a+15 A b^2+9 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {\left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}+\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {\frac {\left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {2 \left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{3 b}+\frac {2 (5 b B-4 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 C \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}\)

Input:

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

Output:

(2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) + (((2*(1 
5*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE 
[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ( 
2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - a*b^2*(15*A + 7*C))*Sqrt[(a + b*Cos[c 
+ d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Co 
s[c + d*x]]))/(3*b) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + 
d*x])/(3*b*d))/(5*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[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 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[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 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[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 3528

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] + Simp[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])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1257\) vs. \(2(247)=494\).

Time = 17.76 (sec) , antiderivative size = 1258, normalized size of antiderivative = 4.88

method	result	size
default	\(\text {Expression too large to display}\)	\(1258\)
parts	\(\text {Expression too large to display}\)	\(1340\)

Input:

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

Output:

2/15*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-24*C*co 
s(1/2*d*x+1/2*c)^7*b^3-20*B*cos(1/2*d*x+1/2*c)^5*b^3+48*C*cos(1/2*d*x+1/2* 
c)^5*b^3+30*B*cos(1/2*d*x+1/2*c)^3*b^3-30*C*cos(1/2*d*x+1/2*c)^3*b^3-10*B* 
cos(1/2*d*x+1/2*c)*b^3+6*C*cos(1/2*d*x+1/2*c)*b^3+15*A*(sin(1/2*d*x+1/2*c) 
^2)^(1/2)*((2*b*cos(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^3-5*B*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2 
*b*cos(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))+8*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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- 
8*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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+9*C*(sin(1/2*d*x+ 
1/2*c)^2)^(1/2)*((2*b*cos(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^3+4*C*cos(1/2*d*x+1/2*c)^5*a*b^2-10* 
B*cos(1/2*d*x+1/2*c)^3*a*b^2+8*C*cos(1/2*d*x+1/2*c)^3*a^2*b-6*C*cos(1/2*d* 
x+1/2*c)^3*a*b^2+10*B*cos(1/2*d*x+1/2*c)*a*b^2-8*C*cos(1/2*d*x+1/2*c)*a^2* 
b+2*C*cos(1/2*d*x+1/2*c)*a*b^2-10*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos 
(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^2+7*C*a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^ 
2+8*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.00 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

-2/45*(sqrt(1/2)*(-16*I*C*a^3 + 20*I*B*a^2*b - 6*I*(5*A + 2*C)*a*b^2 + 15* 
I*B*b^3)*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) + sq 
rt(1/2)*(16*I*C*a^3 - 20*I*B*a^2*b + 6*I*(5*A + 2*C)*a*b^2 - 15*I*B*b^3)*s 
qrt(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(1/2)* 
(-8*I*C*a^2*b + 10*I*B*a*b^2 - 3*I*(5*A + 3*C)*b^3)*sqrt(b)*weierstrassZet 
a(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInvers 
e(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + 
 c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sqrt(1/2)*(8*I*C*a^2*b - 10*I*B*a* 
b^2 + 3*I*(5*A + 3*C)*b^3)*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*(3*C*b^3*cos(d*x + c) - 4*C*a*b^2 + 5*B*b^3)*sqrt(b*cos(d*x 
 + c) + a)*sin(d*x + c))/(b^4*d)
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x))/(cos(c + d*x)*b + a),x)*a + in 
t((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3)/(cos(c + d*x)*b + a),x)*c + i 
nt((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2)/(cos(c + d*x)*b + a),x)*b