\(\int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx\) [329]

Optimal result

Integrand size = 20, antiderivative size = 292 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\frac {a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \arctan \left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{7/2} e}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 \left (a^2-b^2-c^2\right ) e (a+b \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 \left (a^2-b^2-c^2\right )^2 e (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{6 \left (a^2-b^2-c^2\right )^3 e (a+b \cos (d+e x)+c \sin (d+e x))} \] Output:

a*(2*a^2+3*b^2+3*c^2)*arctan((c+(a-b)*tan(1/2*e*x+1/2*d))/(a^2-b^2-c^2)^(1 
/2))/(a^2-b^2-c^2)^(7/2)/e+1/3*(c*cos(e*x+d)-b*sin(e*x+d))/(a^2-b^2-c^2)/e 
/(a+b*cos(e*x+d)+c*sin(e*x+d))^3+5/6*(a*c*cos(e*x+d)-a*b*sin(e*x+d))/(a^2- 
b^2-c^2)^2/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^2+1/6*(c*(11*a^2+4*b^2+4*c^2)*c 
os(e*x+d)-b*(11*a^2+4*b^2+4*c^2)*sin(e*x+d))/(a^2-b^2-c^2)^3/e/(a+b*cos(e* 
x+d)+c*sin(e*x+d))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(606\) vs. \(2(292)=584\).

Time = 1.62 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\frac {\frac {24 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \text {arctanh}\left (\frac {c+(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2+c^2}}\right )}{\left (-a^2+b^2+c^2\right )^{7/2}}+\frac {44 a^5 c+82 a^3 b^2 c+24 a b^4 c+82 a^3 c^3+48 a b^2 c^3+24 a c^5+30 a^2 b c \left (2 a^2+3 \left (b^2+c^2\right )\right ) \cos (d+e x)-6 a c \left (-2 b^4+2 b^2 c^2+4 c^4+a^2 \left (7 b^2+11 c^2\right )\right ) \cos (2 (d+e x))-22 a^2 b^3 c \cos (3 (d+e x))-8 b^5 c \cos (3 (d+e x))-22 a^2 b c^3 \cos (3 (d+e x))-16 b^3 c^3 \cos (3 (d+e x))-8 b c^5 \cos (3 (d+e x))+72 a^4 b^2 \sin (d+e x)-9 a^2 b^4 \sin (d+e x)+12 b^6 \sin (d+e x)+132 a^4 c^2 \sin (d+e x)+72 a^2 b^2 c^2 \sin (d+e x)+36 b^4 c^2 \sin (d+e x)+81 a^2 c^4 \sin (d+e x)+36 b^2 c^4 \sin (d+e x)+12 c^6 \sin (d+e x)+54 a^3 b^3 \sin (2 (d+e x))+6 a b^5 \sin (2 (d+e x))+78 a^3 b c^2 \sin (2 (d+e x))+48 a b^3 c^2 \sin (2 (d+e x))+42 a b c^4 \sin (2 (d+e x))+11 a^2 b^4 \sin (3 (d+e x))+4 b^6 \sin (3 (d+e x))+4 b^4 c^2 \sin (3 (d+e x))-11 a^2 c^4 \sin (3 (d+e x))-4 b^2 c^4 \sin (3 (d+e x))-4 c^6 \sin (3 (d+e x))}{b \left (-a^2+b^2+c^2\right )^3 (a+b \cos (d+e x)+c \sin (d+e x))^3}}{24 e} \] Input:

Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]
 

Output:

((24*a*(2*a^2 + 3*(b^2 + c^2))*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt 
[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(7/2) + (44*a^5*c + 82*a^3*b^2*c + 
 24*a*b^4*c + 82*a^3*c^3 + 48*a*b^2*c^3 + 24*a*c^5 + 30*a^2*b*c*(2*a^2 + 3 
*(b^2 + c^2))*Cos[d + e*x] - 6*a*c*(-2*b^4 + 2*b^2*c^2 + 4*c^4 + a^2*(7*b^ 
2 + 11*c^2))*Cos[2*(d + e*x)] - 22*a^2*b^3*c*Cos[3*(d + e*x)] - 8*b^5*c*Co 
s[3*(d + e*x)] - 22*a^2*b*c^3*Cos[3*(d + e*x)] - 16*b^3*c^3*Cos[3*(d + e*x 
)] - 8*b*c^5*Cos[3*(d + e*x)] + 72*a^4*b^2*Sin[d + e*x] - 9*a^2*b^4*Sin[d 
+ e*x] + 12*b^6*Sin[d + e*x] + 132*a^4*c^2*Sin[d + e*x] + 72*a^2*b^2*c^2*S 
in[d + e*x] + 36*b^4*c^2*Sin[d + e*x] + 81*a^2*c^4*Sin[d + e*x] + 36*b^2*c 
^4*Sin[d + e*x] + 12*c^6*Sin[d + e*x] + 54*a^3*b^3*Sin[2*(d + e*x)] + 6*a* 
b^5*Sin[2*(d + e*x)] + 78*a^3*b*c^2*Sin[2*(d + e*x)] + 48*a*b^3*c^2*Sin[2* 
(d + e*x)] + 42*a*b*c^4*Sin[2*(d + e*x)] + 11*a^2*b^4*Sin[3*(d + e*x)] + 4 
*b^6*Sin[3*(d + e*x)] + 4*b^4*c^2*Sin[3*(d + e*x)] - 11*a^2*c^4*Sin[3*(d + 
 e*x)] - 4*b^2*c^4*Sin[3*(d + e*x)] - 4*c^6*Sin[3*(d + e*x)])/(b*(-a^2 + b 
^2 + c^2)^3*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3))/(24*e)
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3608, 25, 3042, 3635, 25, 3042, 3632, 3042, 3603, 1083, 217}

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.

\(\displaystyle \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4}dx\)

\(\Big \downarrow \) 3608

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3635

\(\displaystyle \frac {\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}-\frac {\int -\frac {2 \left (3 a^2+2 \left (b^2+c^2\right )\right )-5 a b \cos (d+e x)-5 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^2}dx}{2 \left (a^2-b^2-c^2\right )}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 \left (3 a^2+2 \left (b^2+c^2\right )\right )-5 a b \cos (d+e x)-5 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^2}dx}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {2 \left (3 a^2+2 \left (b^2+c^2\right )\right )-5 a b \cos (d+e x)-5 a c \sin (d+e x)}{(a+b \cos (d+e x)+c \sin (d+e x))^2}dx}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3632

\(\displaystyle \frac {\frac {\frac {3 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)}dx}{a^2-b^2-c^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \int \frac {1}{a+b \cos (d+e x)+c \sin (d+e x)}dx}{a^2-b^2-c^2}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3603

\(\displaystyle \frac {\frac {\frac {6 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (d+e x)\right )+2 c \tan \left (\frac {1}{2} (d+e x)\right )+a+b}d\tan \left (\frac {1}{2} (d+e x)\right )}{e \left (a^2-b^2-c^2\right )}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}-\frac {12 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \int \frac {1}{-\left (2 c+2 (a-b) \tan \left (\frac {1}{2} (d+e x)\right )\right )^2-4 \left (a^2-b^2-c^2\right )}d\left (2 c+2 (a-b) \tan \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2-b^2-c^2\right )}}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {6 a \left (2 a^2+3 \left (b^2+c^2\right )\right ) \arctan \left (\frac {2 (a-b) \tan \left (\frac {1}{2} (d+e x)\right )+2 c}{2 \sqrt {a^2-b^2-c^2}}\right )}{e \left (a^2-b^2-c^2\right )^{3/2}}+\frac {c \left (11 a^2+4 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (11 a^2+4 \left (b^2+c^2\right )\right ) \sin (d+e x)}{e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))}}{2 \left (a^2-b^2-c^2\right )}+\frac {5 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^2}}{3 \left (a^2-b^2-c^2\right )}+\frac {c \cos (d+e x)-b \sin (d+e x)}{3 e \left (a^2-b^2-c^2\right ) (a+b \cos (d+e x)+c \sin (d+e x))^3}\)

Input:

Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]
 

Output:

(c*Cos[d + e*x] - b*Sin[d + e*x])/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e* 
x] + c*Sin[d + e*x])^3) + ((5*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(2*(a 
^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + ((6*a*(2*a^2 
+ 3*(b^2 + c^2))*ArcTan[(2*c + 2*(a - b)*Tan[(d + e*x)/2])/(2*Sqrt[a^2 - b 
^2 - c^2])])/((a^2 - b^2 - c^2)^(3/2)*e) + (c*(11*a^2 + 4*(b^2 + c^2))*Cos 
[d + e*x] - b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/((a^2 - b^2 - c^2)*e* 
(a + b*Cos[d + e*x] + c*Sin[d + e*x])))/(2*(a^2 - b^2 - c^2)))/(3*(a^2 - b 
^2 - c^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3603

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

rule 3608

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 
1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c 
*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x 
] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && 
NeQ[n, -3/2]
 

rule 3632

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

rule 3635

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) 
^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) 
*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( 
a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 
2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] 
/; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 
 0] && NeQ[n, -2]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.47 (sec) , antiderivative size = 1655, normalized size of antiderivative = 5.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1655\)
derivativedivides	\(\text {Expression too large to display}\)	\(1656\)
default	\(\text {Expression too large to display}\)	\(1656\)

Input:

int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x,method=_RETURNVERBOSE)
 

Output:

1/3*I*(-4*c^5-11*a^2*c^3+60*I*a^3*c^2*exp(I*(e*x+d))+15*I*a*c^4*exp(I*(e*x 
+d))-4*I*b^5+33*a^2*b^2*c+12*I*b*c^4+8*I*b^3*c^2+30*a*b^3*c*exp(I*(e*x+d)) 
+120*a^3*b*c*exp(I*(e*x+d))+30*a*b*c^3*exp(I*(e*x+d))+6*I*a^3*c^2*exp(5*I* 
(e*x+d))+9*I*a*c^4*exp(5*I*(e*x+d))-24*I*a*c^4*exp(3*I*(e*x+d))+102*a^4*c* 
exp(2*I*(e*x+d))-12*I*b^5*exp(2*I*(e*x+d))+33*I*a^2*b*c^2-60*I*a^3*b^2*exp 
(I*(e*x+d))-15*I*a*b^4*exp(I*(e*x+d))+12*b^4*c+8*b^2*c^3-12*a^3*b*c*exp(5* 
I*(e*x+d))+36*a^2*b^2*c*exp(2*I*(e*x+d))-18*a*b^3*c*exp(5*I*(e*x+d))-11*I* 
a^2*b^3+24*b^2*c^3*exp(2*I*(e*x+d))+12*b^4*c*exp(2*I*(e*x+d))-36*I*a^2*b*c 
^2*exp(2*I*(e*x+d))-45*I*a^2*b*c^2*exp(4*I*(e*x+d))-48*I*a*b^2*c^2*exp(3*I 
*(e*x+d))-30*I*a^4*b*exp(4*I*(e*x+d))-9*I*a*b^4*exp(5*I*(e*x+d))-45*I*a^2* 
b^3*exp(4*I*(e*x+d))-82*I*a^3*b^2*exp(3*I*(e*x+d))-82*I*a^3*c^2*exp(3*I*(e 
*x+d))-44*I*exp(3*I*(e*x+d))*a^5-24*I*a*b^4*exp(3*I*(e*x+d))-36*I*a^2*b^3* 
exp(2*I*(e*x+d))-24*I*b^3*c^2*exp(2*I*(e*x+d))-102*I*a^4*b*exp(2*I*(e*x+d) 
)-12*I*b*c^4*exp(2*I*(e*x+d))-30*a^4*c*exp(4*I*(e*x+d))-45*a^2*c^3*exp(4*I 
*(e*x+d))+36*a^2*c^3*exp(2*I*(e*x+d))-45*a^2*b^2*c*exp(4*I*(e*x+d))-18*a*b 
*c^3*exp(5*I*(e*x+d))-6*I*a^3*b^2*exp(5*I*(e*x+d))+12*c^5*exp(2*I*(e*x+d)) 
)/(c*exp(2*I*(e*x+d))+I*b*exp(2*I*(e*x+d))-c+2*I*a*exp(I*(e*x+d))+I*b)^3/( 
-a^2+b^2+c^2)^3/e-1/(-a^2+b^2+c^2)^(1/2)*a^3/(a^2-b^2-c^2)^3/e*ln(exp(I*(e 
*x+d))+(I*a*c*(-a^2+b^2+c^2)^(1/2)+I*a^2*b-I*b^3-I*b*c^2+b*a*(-a^2+b^2+c^2 
)^(1/2)-a^2*c+b^2*c+c^3)/(b^2+c^2)/(-a^2+b^2+c^2)^(1/2))-3/2/(-a^2+b^2+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1940 vs. \(2 (284) = 568\).

Time = 0.29 (sec) , antiderivative size = 4069, normalized size of antiderivative = 13.93 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,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(c^2+b^2-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. 2558 vs. \(2 (284) = 568\).

Time = 0.25 (sec) , antiderivative size = 2558, normalized size of antiderivative = 8.76 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*(3*(2*a^3 + 3*a*b^2 + 3*a*c^2)*(pi*floor(1/2*(e*x + d)/pi + 1/2)*sgn( 
-2*a + 2*b) + arctan(-(a*tan(1/2*e*x + 1/2*d) - b*tan(1/2*e*x + 1/2*d) + c 
)/sqrt(a^2 - b^2 - c^2)))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 
+ 6*a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 - c^6)*sqrt(a^2 - b^2 
- c^2)) + (18*a^7*b*tan(1/2*e*x + 1/2*d)^5 - 81*a^6*b^2*tan(1/2*e*x + 1/2* 
d)^5 + 141*a^5*b^3*tan(1/2*e*x + 1/2*d)^5 - 120*a^4*b^4*tan(1/2*e*x + 1/2* 
d)^5 + 60*a^3*b^5*tan(1/2*e*x + 1/2*d)^5 - 33*a^2*b^6*tan(1/2*e*x + 1/2*d) 
^5 + 21*a*b^7*tan(1/2*e*x + 1/2*d)^5 - 6*b^8*tan(1/2*e*x + 1/2*d)^5 - 27*a 
^6*c^2*tan(1/2*e*x + 1/2*d)^5 + 81*a^5*b*c^2*tan(1/2*e*x + 1/2*d)^5 - 72*a 
^4*b^2*c^2*tan(1/2*e*x + 1/2*d)^5 + 18*a^3*b^3*c^2*tan(1/2*e*x + 1/2*d)^5 
- 27*a^2*b^4*c^2*tan(1/2*e*x + 1/2*d)^5 + 45*a*b^5*c^2*tan(1/2*e*x + 1/2*d 
)^5 - 18*b^6*c^2*tan(1/2*e*x + 1/2*d)^5 + 18*a^4*c^4*tan(1/2*e*x + 1/2*d)^ 
5 - 36*a^3*b*c^4*tan(1/2*e*x + 1/2*d)^5 + 36*a*b^3*c^4*tan(1/2*e*x + 1/2*d 
)^5 - 18*b^4*c^4*tan(1/2*e*x + 1/2*d)^5 - 6*a^2*c^6*tan(1/2*e*x + 1/2*d)^5 
 + 12*a*b*c^6*tan(1/2*e*x + 1/2*d)^5 - 6*b^2*c^6*tan(1/2*e*x + 1/2*d)^5 - 
18*a^7*c*tan(1/2*e*x + 1/2*d)^4 + 108*a^6*b*c*tan(1/2*e*x + 1/2*d)^4 - 261 
*a^5*b^2*c*tan(1/2*e*x + 1/2*d)^4 + 336*a^4*b^3*c*tan(1/2*e*x + 1/2*d)^4 - 
 264*a^3*b^4*c*tan(1/2*e*x + 1/2*d)^4 + 144*a^2*b^5*c*tan(1/2*e*x + 1/2*d) 
^4 - 57*a*b^6*c*tan(1/2*e*x + 1/2*d)^4 + 12*b^7*c*tan(1/2*e*x + 1/2*d)^4 - 
 81*a^5*c^3*tan(1/2*e*x + 1/2*d)^4 + 216*a^4*b*c^3*tan(1/2*e*x + 1/2*d)...
 

Mupad [B] (verification not implemented)

Time = 17.75 (sec) , antiderivative size = 1946, normalized size of antiderivative = 6.66 \[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:

int(1/(a + b*cos(d + e*x) + c*sin(d + e*x))^4,x)
 

Output:

(a*atanh((a*(2*a^2 + 3*b^2 + 3*c^2)*(2*b^6*c - 2*a^6*c + 2*c^7 - 6*a^2*c^5 
 + 6*a^4*c^3 + 6*b^2*c^5 + 6*b^4*c^3 - 6*a^2*b^4*c + 6*a^4*b^2*c - 12*a^2* 
b^2*c^3))/(2*(b^2 - a^2 + c^2)^(7/2)*(3*a*b^2 + 3*a*c^2 + 2*a^3)) + (a*tan 
(d/2 + (e*x)/2)*(2*a - 2*b)*(2*a^2 + 3*b^2 + 3*c^2)*(b^6 - a^6 + c^6 - 3*a 
^2*b^4 + 3*a^4*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2 
*b^2*c^2))/(2*(b^2 - a^2 + c^2)^(7/2)*(3*a*b^2 + 3*a*c^2 + 2*a^3)))*(2*a^2 
 + 3*b^2 + 3*c^2))/(e*(b^2 - a^2 + c^2)^(7/2)) - ((18*a^7*c + 2*a^3*c^5 - 
5*a^5*c^3 + 6*a*b^2*c^5 + 21*a*b^4*c^3 - 12*a^3*b^4*c - 21*a^5*b^2*c - 16* 
a^3*b^2*c^3 + 15*a*b^6*c)/(3*(a - b)^3*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*a^ 
4*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) + 
(tan(d/2 + (e*x)/2)*(2*b^8 - 6*a^7*b - 5*a*b^7 + 5*a^2*b^6 + 4*a^3*b^5 - 1 
6*a^4*b^4 + 7*a^5*b^3 + 9*a^6*b^2 + 2*a^2*c^6 - 4*a^4*c^4 + 27*a^6*c^2 + 2 
*b^2*c^6 + 6*b^4*c^4 + 6*b^6*c^2 + 18*a*b^3*c^4 + 9*a*b^5*c^2 - 14*a^3*b*c 
^4 - 9*a^5*b*c^2 - 6*a^2*b^2*c^4 - 3*a^2*b^4*c^2 - 30*a^4*b^2*c^2 + 4*a*b* 
c^6))/((a - b)^3*(b^6 - a^6 + c^6 - 3*a^2*b^4 + 3*a^4*b^2 - 3*a^2*c^4 + 3* 
a^4*c^2 + 3*b^2*c^4 + 3*b^4*c^2 - 6*a^2*b^2*c^2)) + (tan(d/2 + (e*x)/2)^4* 
(6*a^6*c + 4*b^6*c + 4*c^7 - 12*a^2*c^5 + 27*a^4*c^3 + 12*b^2*c^5 + 12*b^4 
*c^3 - 15*a*b^3*c^3 + 33*a^2*b^4*c - 45*a^3*b*c^3 - 55*a^3*b^3*c + 57*a^4* 
b^2*c + 21*a^2*b^2*c^3 - 15*a*b^5*c - 30*a^5*b*c))/((a - b)^2*(b^6 - a^6 + 
 c^6 - 3*a^2*b^4 + 3*a^4*b^2 - 3*a^2*c^4 + 3*a^4*c^2 + 3*b^2*c^4 + 3*b^...
 

Reduce [F]

\[ \int \frac {1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx=\int \frac {1}{\left (a +b \cos \left (e x +d \right )+c \sin \left (e x +d \right )\right )^{4}}d x \] Input:

int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x)
 

Output:

int(1/(a+b*cos(e*x+d)+c*sin(e*x+d))^4,x)