\(\int \frac {\cos ^3(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [977]

Optimal result

Integrand size = 41, antiderivative size = 279 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=-\frac {\left (8 a^3 b B+4 a b^3 B-8 a^4 C-4 a^2 b^2 (2 A+C)-b^4 (4 A+3 C)\right ) x}{8 b^5}-\frac {2 a^3 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^5 \sqrt {a+b} d}+\frac {\left (3 a^2 b B+2 b^3 B-3 a^3 C-a b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d}+\frac {\left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d} \] Output:

-1/8*(8*B*a^3*b+4*B*a*b^3-8*a^4*C-4*a^2*b^2*(2*A+C)-b^4*(4*A+3*C))*x/b^5-2 
*a^3*(A*b^2-a*(B*b-C*a))*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2) 
)/(a-b)^(1/2)/b^5/(a+b)^(1/2)/d+1/3*(3*B*a^2*b+2*B*b^3-3*a^3*C-a*b^2*(3*A+ 
2*C))*sin(d*x+c)/b^4/d+1/8*(4*A*b^2-4*B*a*b+4*C*a^2+3*C*b^2)*cos(d*x+c)*si 
n(d*x+c)/b^3/d+1/3*(B*b-C*a)*cos(d*x+c)^2*sin(d*x+c)/b^2/d+1/4*C*cos(d*x+c 
)^3*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 2.71 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {12 \left (-8 a^3 b B-4 a b^3 B+8 a^4 C+4 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) (c+d x)+\frac {192 a^3 \left (A b^2+a (-b B+a C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+24 b \left (4 a^2 b B+3 b^3 B-4 a^3 C-a b^2 (4 A+3 C)\right ) \sin (c+d x)+24 b^2 \left (A b^2-a b B+a^2 C+b^2 C\right ) \sin (2 (c+d x))+8 b^3 (b B-a C) \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 b^5 d} \] Input:

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

Output:

(12*(-8*a^3*b*B - 4*a*b^3*B + 8*a^4*C + 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3 
*C))*(c + d*x) + (192*a^3*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[ 
(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 24*b*(4*a^2*b*B + 3*b^ 
3*B - 4*a^3*C - a*b^2*(4*A + 3*C))*Sin[c + d*x] + 24*b^2*(A*b^2 - a*b*B + 
a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b^3*(b*B - a*C)*Sin[3*(c + d*x)] + 3*b 
^4*C*Sin[4*(c + d*x)])/(96*b^5*d)
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 3528, 3042, 3528, 3042, 3528, 3042, 3502, 27, 3042, 3214, 3042, 3138, 218}

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.

\(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\frac {\int \frac {8 \left (-3 C a^3+3 b B a^2-b^2 (3 A+2 C) a+2 b^3 B\right ) \cos ^2(c+d x)+b \left (-4 C a^2+4 b B a+12 A b^2+9 b^2 C\right ) \cos (c+d x)+3 a \left (4 C a^2-4 b B a+4 A b^2+3 b^2 C\right )}{a+b \cos (c+d x)}dx}{2 b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )}{2 b d}}{3 b}+\frac {4 (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {8 \left (-3 C a^3+3 b B a^2-b^2 (3 A+2 C) a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-4 C a^2+4 b B a+12 A b^2+9 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (4 C a^2-4 b B a+4 A b^2+3 b^2 C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )}{2 b d}}{3 b}+\frac {4 (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 b d}\)

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3214

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {16 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}-\frac {x \left (-8 a^4 C+8 a^3 b B-4 a^2 b^2 (2 A+C)+4 a b^3 B-b^4 (4 A+3 C)\right )}{b}\right )}{b}+\frac {8 \sin (c+d x) \left (-3 a^3 C+3 a^2 b B-a b^2 (3 A+2 C)+2 b^3 B\right )}{b d}}{2 b}+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )}{2 b d}}{3 b}+\frac {4 (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 b d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )}{2 b d}+\frac {\frac {8 \sin (c+d x) \left (-3 a^3 C+3 a^2 b B-a b^2 (3 A+2 C)+2 b^3 B\right )}{b d}+\frac {3 \left (-\frac {16 a^3 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (-8 a^4 C+8 a^3 b B-4 a^2 b^2 (2 A+C)+4 a b^3 B-b^4 (4 A+3 C)\right )}{b}\right )}{b}}{2 b}}{3 b}+\frac {4 (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d}}{4 b}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 b d}\)

Input:

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

Output:

(C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d) + ((4*(b*B - a*C)*Cos[c + d*x]^2*S 
in[c + d*x])/(3*b*d) + ((3*(4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Cos[c + 
 d*x]*Sin[c + d*x])/(2*b*d) + ((3*(-(((8*a^3*b*B + 4*a*b^3*B - 8*a^4*C - 4 
*a^2*b^2*(2*A + C) - b^4*(4*A + 3*C))*x)/b) - (16*a^3*(A*b^2 - a*(b*B - a* 
C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqr 
t[a + b]*d)))/b + (8*(3*a^2*b*B + 2*b^3*B - 3*a^3*C - a*b^2*(3*A + 2*C))*S 
in[c + d*x])/(b*d))/(2*b))/(3*b))/(4*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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 3042

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

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 3214

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d   Int[1/(c + d 
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 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 [A] (verified)

Time = 1.41 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.63

method	result	size
derivativedivides	\(\frac {-\frac {2 a^{3} \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (-a A \,b^{3}-\frac {1}{2} A \,b^{4}+B \,a^{2} b^{2}+\frac {1}{2} B a \,b^{3}+B \,b^{4}-a^{3} b C -\frac {1}{2} C \,a^{2} b^{2}-C a \,b^{3}-\frac {5}{8} C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-3 a^{3} b C -3 a A \,b^{3}+\frac {1}{2} B a \,b^{3}+3 B \,a^{2} b^{2}+\frac {5}{3} B \,b^{4}-\frac {1}{2} A \,b^{4}-\frac {5}{3} C a \,b^{3}-\frac {1}{2} C \,a^{2} b^{2}+\frac {3}{8} C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {1}{2} A \,b^{4}-\frac {1}{2} B a \,b^{3}+\frac {1}{2} C \,a^{2} b^{2}-\frac {3}{8} C \,b^{4}-3 a^{3} b C -3 a A \,b^{3}+3 B \,a^{2} b^{2}+\frac {5}{3} B \,b^{4}-\frac {5}{3} C a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} A \,b^{4}-\frac {1}{2} B a \,b^{3}+\frac {1}{2} C \,a^{2} b^{2}+\frac {5}{8} C \,b^{4}-a A \,b^{3}+B \,a^{2} b^{2}+B \,b^{4}-a^{3} b C -C a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (8 A \,a^{2} b^{2}+4 A \,b^{4}-8 B \,a^{3} b -4 B a \,b^{3}+8 a^{4} C +4 C \,a^{2} b^{2}+3 C \,b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{5}}}{d}\)	\(456\)
default	\(\frac {-\frac {2 a^{3} \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (-a A \,b^{3}-\frac {1}{2} A \,b^{4}+B \,a^{2} b^{2}+\frac {1}{2} B a \,b^{3}+B \,b^{4}-a^{3} b C -\frac {1}{2} C \,a^{2} b^{2}-C a \,b^{3}-\frac {5}{8} C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-3 a^{3} b C -3 a A \,b^{3}+\frac {1}{2} B a \,b^{3}+3 B \,a^{2} b^{2}+\frac {5}{3} B \,b^{4}-\frac {1}{2} A \,b^{4}-\frac {5}{3} C a \,b^{3}-\frac {1}{2} C \,a^{2} b^{2}+\frac {3}{8} C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {1}{2} A \,b^{4}-\frac {1}{2} B a \,b^{3}+\frac {1}{2} C \,a^{2} b^{2}-\frac {3}{8} C \,b^{4}-3 a^{3} b C -3 a A \,b^{3}+3 B \,a^{2} b^{2}+\frac {5}{3} B \,b^{4}-\frac {5}{3} C a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} A \,b^{4}-\frac {1}{2} B a \,b^{3}+\frac {1}{2} C \,a^{2} b^{2}+\frac {5}{8} C \,b^{4}-a A \,b^{3}+B \,a^{2} b^{2}+B \,b^{4}-a^{3} b C -C a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (8 A \,a^{2} b^{2}+4 A \,b^{4}-8 B \,a^{3} b -4 B a \,b^{3}+8 a^{4} C +4 C \,a^{2} b^{2}+3 C \,b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{5}}}{d}\)	\(456\)
risch	\(\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B}{8 b d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B}{8 b d}+\frac {\sin \left (3 d x +3 c \right ) B}{12 b d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 b d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 b d}+\frac {x A \,a^{2}}{b^{3}}-\frac {x B \,a^{3}}{b^{4}}-\frac {x B a}{2 b^{2}}+\frac {x \,a^{4} C}{b^{5}}+\frac {x C \,a^{2}}{2 b^{3}}+\frac {C \sin \left (4 d x +4 c \right )}{32 b d}+\frac {3 C x}{8 b}+\frac {i {\mathrm e}^{i \left (d x +c \right )} a A}{2 b^{2} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2}}{2 b^{3} d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 b^{4} d}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C a}{8 b^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} a A}{2 b^{2} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2}}{2 b^{3} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 b^{4} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C a}{8 b^{2} d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, d \,b^{4}}-\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{5}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, d \,b^{4}}+\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{5}}-\frac {\sin \left (3 d x +3 c \right ) C a}{12 b^{2} d}-\frac {\sin \left (2 d x +2 c \right ) B a}{4 b^{2} d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{4 b^{3} d}+\frac {x A}{2 b}\)	\(846\)

Input:

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

Output:

1/d*(-2*a^3*(A*b^2-B*a*b+C*a^2)/b^5/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1 
/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))+2/b^5*(((-a*A*b^3-1/2*A*b^4+B*a^2*b^2+1 
/2*B*a*b^3+B*b^4-a^3*b*C-1/2*C*a^2*b^2-C*a*b^3-5/8*C*b^4)*tan(1/2*d*x+1/2* 
c)^7+(-3*a^3*b*C-3*a*A*b^3+1/2*B*a*b^3+3*B*a^2*b^2+5/3*B*b^4-1/2*A*b^4-5/3 
*C*a*b^3-1/2*C*a^2*b^2+3/8*C*b^4)*tan(1/2*d*x+1/2*c)^5+(1/2*A*b^4-1/2*B*a* 
b^3+1/2*C*a^2*b^2-3/8*C*b^4-3*a^3*b*C-3*a*A*b^3+3*B*a^2*b^2+5/3*B*b^4-5/3* 
C*a*b^3)*tan(1/2*d*x+1/2*c)^3+(1/2*A*b^4-1/2*B*a*b^3+1/2*C*a^2*b^2+5/8*C*b 
^4-a*A*b^3+B*a^2*b^2+B*b^4-a^3*b*C-C*a*b^3)*tan(1/2*d*x+1/2*c))/(1+tan(1/2 
*d*x+1/2*c)^2)^4+1/8*(8*A*a^2*b^2+4*A*b^4-8*B*a^3*b-4*B*a*b^3+8*C*a^4+4*C* 
a^2*b^2+3*C*b^4)*arctan(tan(1/2*d*x+1/2*c))))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.78 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/24*(3*(8*C*a^6 - 8*B*a^5*b + 4*(2*A - C)*a^4*b^2 + 4*B*a^3*b^3 - (4*A + 
 C)*a^2*b^4 + 4*B*a*b^5 - (4*A + 3*C)*b^6)*d*x - 12*(C*a^5 - B*a^4*b + A*a 
^3*b^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + 
 c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2 
)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - (24*C*a^5*b - 24*B*a^ 
4*b^2 + 8*(3*A - C)*a^3*b^3 + 8*B*a^2*b^4 - 8*(3*A + 2*C)*a*b^5 + 16*B*b^6 
 - 6*(C*a^2*b^4 - C*b^6)*cos(d*x + c)^3 + 8*(C*a^3*b^3 - B*a^2*b^4 - C*a*b 
^5 + B*b^6)*cos(d*x + c)^2 - 3*(4*C*a^4*b^2 - 4*B*a^3*b^3 + (4*A - C)*a^2* 
b^4 + 4*B*a*b^5 - (4*A + 3*C)*b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^5 - 
 b^7)*d), 1/24*(3*(8*C*a^6 - 8*B*a^5*b + 4*(2*A - C)*a^4*b^2 + 4*B*a^3*b^3 
 - (4*A + C)*a^2*b^4 + 4*B*a*b^5 - (4*A + 3*C)*b^6)*d*x - 24*(C*a^5 - B*a^ 
4*b + A*a^3*b^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - 
b^2)*sin(d*x + c))) - (24*C*a^5*b - 24*B*a^4*b^2 + 8*(3*A - C)*a^3*b^3 + 8 
*B*a^2*b^4 - 8*(3*A + 2*C)*a*b^5 + 16*B*b^6 - 6*(C*a^2*b^4 - C*b^6)*cos(d* 
x + c)^3 + 8*(C*a^3*b^3 - B*a^2*b^4 - C*a*b^5 + B*b^6)*cos(d*x + c)^2 - 3* 
(4*C*a^4*b^2 - 4*B*a^3*b^3 + (4*A - C)*a^2*b^4 + 4*B*a*b^5 - (4*A + 3*C)*b 
^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^5 - b^7)*d)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (257) = 514\).

Time = 0.15 (sec) , antiderivative size = 801, normalized size of antiderivative = 2.87 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(3*(8*C*a^4 - 8*B*a^3*b + 8*A*a^2*b^2 + 4*C*a^2*b^2 - 4*B*a*b^3 + 4*A 
*b^4 + 3*C*b^4)*(d*x + c)/b^5 + 48*(C*a^5 - B*a^4*b + A*a^3*b^2)*(pi*floor 
(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) 
 - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^5) - 2*(24 
*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*C*a 
^2*b*tan(1/2*d*x + 1/2*c)^7 + 24*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 12*B*a*b 
^2*tan(1/2*d*x + 1/2*c)^7 + 24*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^3*t 
an(1/2*d*x + 1/2*c)^7 - 24*B*b^3*tan(1/2*d*x + 1/2*c)^7 + 15*C*b^3*tan(1/2 
*d*x + 1/2*c)^7 + 72*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 72*B*a^2*b*tan(1/2*d*x 
 + 1/2*c)^5 + 12*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 72*A*a*b^2*tan(1/2*d*x + 
 1/2*c)^5 - 12*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 40*C*a*b^2*tan(1/2*d*x + 1 
/2*c)^5 + 12*A*b^3*tan(1/2*d*x + 1/2*c)^5 - 40*B*b^3*tan(1/2*d*x + 1/2*c)^ 
5 - 9*C*b^3*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 72* 
B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 72*A* 
a*b^2*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 40*C*a* 
b^2*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^3*tan(1/2*d*x + 1/2*c)^3 - 40*B*b^3*ta 
n(1/2*d*x + 1/2*c)^3 + 9*C*b^3*tan(1/2*d*x + 1/2*c)^3 + 24*C*a^3*tan(1/2*d 
*x + 1/2*c) - 24*B*a^2*b*tan(1/2*d*x + 1/2*c) - 12*C*a^2*b*tan(1/2*d*x + 1 
/2*c) + 24*A*a*b^2*tan(1/2*d*x + 1/2*c) + 12*B*a*b^2*tan(1/2*d*x + 1/2*c) 
+ 24*C*a*b^2*tan(1/2*d*x + 1/2*c) - 12*A*b^3*tan(1/2*d*x + 1/2*c) - 24*...
 

Mupad [B] (verification not implemented)

Time = 6.26 (sec) , antiderivative size = 9661, normalized size of antiderivative = 34.63 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

- ((tan(c/2 + (d*x)/2)^7*(4*A*b^3 - 8*B*b^3 + 8*C*a^3 + 5*C*b^3 + 8*A*a*b^ 
2 - 4*B*a*b^2 - 8*B*a^2*b + 8*C*a*b^2 + 4*C*a^2*b))/(4*b^4) + (tan(c/2 + ( 
d*x)/2)^3*(72*C*a^3 - 40*B*b^3 - 12*A*b^3 + 9*C*b^3 + 72*A*a*b^2 + 12*B*a* 
b^2 - 72*B*a^2*b + 40*C*a*b^2 - 12*C*a^2*b))/(12*b^4) + (tan(c/2 + (d*x)/2 
)^5*(12*A*b^3 - 40*B*b^3 + 72*C*a^3 - 9*C*b^3 + 72*A*a*b^2 - 12*B*a*b^2 - 
72*B*a^2*b + 40*C*a*b^2 + 12*C*a^2*b))/(12*b^4) - (tan(c/2 + (d*x)/2)*(4*A 
*b^3 + 8*B*b^3 - 8*C*a^3 + 5*C*b^3 - 8*A*a*b^2 - 4*B*a*b^2 + 8*B*a^2*b - 8 
*C*a*b^2 + 4*C*a^2*b))/(4*b^4))/(d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + ( 
d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) - (atan((( 
((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b 
^10 - 27*C^2*a*b^10 + 256*C^2*a^10*b + 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8 + 
 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7*b^4 + 1 
6*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B 
^2*a^6*b^5 - 256*B^2*a^7*b^4 + 256*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C^2* 
a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 256*C^2*a^6 
*b^5 - 256*C^2*a^7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 
 32*A*B*a*b^10 - 72*A*C*a*b^10 - 24*B*C*a*b^10 + 256*B*C*a^10*b + 96*A*B*a 
^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 + 512*A*B*a^6 
*b^5 - 512*A*B*a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b 
^8 + 368*A*C*a^4*b^7 - 464*A*C*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*a^7*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {-48 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) a^{5} c -6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{4} c +6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{6} c +12 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b^{2} c +3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{4} c -15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{6} c +8 \sin \left (d x +c \right )^{3} a^{3} b^{3} c -8 \sin \left (d x +c \right )^{3} a^{2} b^{5}-8 \sin \left (d x +c \right )^{3} a \,b^{5} c +8 \sin \left (d x +c \right )^{3} b^{7}-24 \sin \left (d x +c \right ) a^{5} b c +24 \sin \left (d x +c \right ) a^{2} b^{5}+24 \sin \left (d x +c \right ) a \,b^{5} c -24 \sin \left (d x +c \right ) b^{7}+24 a^{6} c^{2}+24 a^{6} c d x -12 a^{4} b^{2} c^{2}-12 a^{4} b^{2} c d x -3 a^{2} b^{4} c^{2}-3 a^{2} b^{4} c d x -9 b^{6} c^{2}-9 b^{6} c d x}{24 b^{5} d \left (a^{2}-b^{2}\right )} \] Input:

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

Output:

( - 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq 
rt(a**2 - b**2))*a**5*c - 6*cos(c + d*x)*sin(c + d*x)**3*a**2*b**4*c + 6*c 
os(c + d*x)*sin(c + d*x)**3*b**6*c + 12*cos(c + d*x)*sin(c + d*x)*a**4*b** 
2*c + 3*cos(c + d*x)*sin(c + d*x)*a**2*b**4*c - 15*cos(c + d*x)*sin(c + d* 
x)*b**6*c + 8*sin(c + d*x)**3*a**3*b**3*c - 8*sin(c + d*x)**3*a**2*b**5 - 
8*sin(c + d*x)**3*a*b**5*c + 8*sin(c + d*x)**3*b**7 - 24*sin(c + d*x)*a**5 
*b*c + 24*sin(c + d*x)*a**2*b**5 + 24*sin(c + d*x)*a*b**5*c - 24*sin(c + d 
*x)*b**7 + 24*a**6*c**2 + 24*a**6*c*d*x - 12*a**4*b**2*c**2 - 12*a**4*b**2 
*c*d*x - 3*a**2*b**4*c**2 - 3*a**2*b**4*c*d*x - 9*b**6*c**2 - 9*b**6*c*d*x 
)/(24*b**5*d*(a**2 - b**2))