\(\int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx\) [508]

Optimal result

Integrand size = 31, antiderivative size = 258 \[ \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx=\frac {\left (2 a^3 A+3 a A c^2-4 a^2 c C-c^3 C\right ) \arctan \left (\frac {c+a \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-c^2}}\right )}{\left (a^2-c^2\right )^{7/2} e}-\frac {B}{3 c e (a+c \sin (d+e x))^3}+\frac {(A c-a C) \cos (d+e x)}{3 \left (a^2-c^2\right ) e (a+c \sin (d+e x))^3}+\frac {\left (5 a A c-2 a^2 C-3 c^2 C\right ) \cos (d+e x)}{6 \left (a^2-c^2\right )^2 e (a+c \sin (d+e x))^2}+\frac {\left (11 a^2 A c+4 A c^3-2 a^3 C-13 a c^2 C\right ) \cos (d+e x)}{6 \left (a^2-c^2\right )^3 e (a+c \sin (d+e x))} \] Output:

(2*A*a^3+3*A*a*c^2-4*C*a^2*c-C*c^3)*arctan((c+a*tan(1/2*e*x+1/2*d))/(a^2-c 
^2)^(1/2))/(a^2-c^2)^(7/2)/e-1/3*B/c/e/(a+c*sin(e*x+d))^3+1/3*(A*c-C*a)*co 
s(e*x+d)/(a^2-c^2)/e/(a+c*sin(e*x+d))^3+1/6*(5*A*a*c-2*C*a^2-3*C*c^2)*cos( 
e*x+d)/(a^2-c^2)^2/e/(a+c*sin(e*x+d))^2+1/6*(11*A*a^2*c+4*A*c^3-2*C*a^3-13 
*C*a*c^2)*cos(e*x+d)/(a^2-c^2)^3/e/(a+c*sin(e*x+d))
 

Mathematica [A] (verified)

Time = 2.41 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx=\frac {\frac {6 \left (2 a^3 A+3 a A c^2-4 a^2 c C-c^3 C\right ) \arctan \left (\frac {c+a \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-c^2}}\right )}{\left (a^2-c^2\right )^{7/2}}+\frac {2 B \left (-a^2+c^2\right )+2 c (A c-a C) \cos (d+e x)}{(a-c) c (a+c) (a+c \sin (d+e x))^3}+\frac {\left (5 a A c-2 a^2 C-3 c^2 C\right ) \cos (d+e x)}{(a-c)^2 (a+c)^2 (a+c \sin (d+e x))^2}+\frac {\left (11 a^2 A c+4 A c^3-2 a^3 C-13 a c^2 C\right ) \cos (d+e x)}{(a-c)^3 (a+c)^3 (a+c \sin (d+e x))}}{6 e} \] Input:

Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^4,x]
 

Output:

((6*(2*a^3*A + 3*a*A*c^2 - 4*a^2*c*C - c^3*C)*ArcTan[(c + a*Tan[(d + e*x)/ 
2])/Sqrt[a^2 - c^2]])/(a^2 - c^2)^(7/2) + (2*B*(-a^2 + c^2) + 2*c*(A*c - a 
*C)*Cos[d + e*x])/((a - c)*c*(a + c)*(a + c*Sin[d + e*x])^3) + ((5*a*A*c - 
 2*a^2*C - 3*c^2*C)*Cos[d + e*x])/((a - c)^2*(a + c)^2*(a + c*Sin[d + e*x] 
)^2) + ((11*a^2*A*c + 4*A*c^3 - 2*a^3*C - 13*a*c^2*C)*Cos[d + e*x])/((a - 
c)^3*(a + c)^3*(a + c*Sin[d + e*x])))/(6*e)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4876, 3042, 3147, 17, 3233, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3139, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4}dx\)

\(\Big \downarrow \) 4876

\(\displaystyle \int \frac {A+C \sin (d+e x)}{(a+c \sin (d+e x))^4}dx+B \int \frac {\cos (d+e x)}{(a+c \sin (d+e x))^4}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+C \sin (d+e x)}{(a+c \sin (d+e x))^4}dx+B \int \frac {\cos (d+e x)}{(a+c \sin (d+e x))^4}dx\)

\(\Big \downarrow \) 3147

\(\displaystyle \int \frac {A+C \sin (d+e x)}{(a+c \sin (d+e x))^4}dx+\frac {B \int \frac {1}{(a+c \sin (d+e x))^4}d(c \sin (d+e x))}{c e}\)

\(\Big \downarrow \) 17

\(\displaystyle \int \frac {A+C \sin (d+e x)}{(a+c \sin (d+e x))^4}dx-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {\frac {\left (-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right ) \cos (d+e x)}{e \left (a^2-c^2\right ) (a+c \sin (d+e x))}-\frac {\int -\frac {3 \left (2 A a^3-4 c C a^2+3 A c^2 a-c^3 C\right )}{a+c \sin (d+e x)}dx}{a^2-c^2}}{2 \left (a^2-c^2\right )}+\frac {\left (-2 a^2 C+5 a A c-3 c^2 C\right ) \cos (d+e x)}{2 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^2}}{3 \left (a^2-c^2\right )}+\frac {(A c-a C) \cos (d+e x)}{3 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^3}-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right ) \int \frac {1}{a+c \sin (d+e x)}dx}{a^2-c^2}+\frac {\left (-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right ) \cos (d+e x)}{e \left (a^2-c^2\right ) (a+c \sin (d+e x))}}{2 \left (a^2-c^2\right )}+\frac {\left (-2 a^2 C+5 a A c-3 c^2 C\right ) \cos (d+e x)}{2 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^2}}{3 \left (a^2-c^2\right )}+\frac {(A c-a C) \cos (d+e x)}{3 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^3}-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right ) \int \frac {1}{a+c \sin (d+e x)}dx}{a^2-c^2}+\frac {\left (-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right ) \cos (d+e x)}{e \left (a^2-c^2\right ) (a+c \sin (d+e x))}}{2 \left (a^2-c^2\right )}+\frac {\left (-2 a^2 C+5 a A c-3 c^2 C\right ) \cos (d+e x)}{2 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^2}}{3 \left (a^2-c^2\right )}+\frac {(A c-a C) \cos (d+e x)}{3 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^3}-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {\frac {\frac {6 \left (2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (d+e x)\right )+2 c \tan \left (\frac {1}{2} (d+e x)\right )+a}d\tan \left (\frac {1}{2} (d+e x)\right )}{e \left (a^2-c^2\right )}+\frac {\left (-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right ) \cos (d+e x)}{e \left (a^2-c^2\right ) (a+c \sin (d+e x))}}{2 \left (a^2-c^2\right )}+\frac {\left (-2 a^2 C+5 a A c-3 c^2 C\right ) \cos (d+e x)}{2 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^2}}{3 \left (a^2-c^2\right )}+\frac {(A c-a C) \cos (d+e x)}{3 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^3}-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {\frac {\left (-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right ) \cos (d+e x)}{e \left (a^2-c^2\right ) (a+c \sin (d+e x))}-\frac {12 \left (2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right ) \int \frac {1}{-\left (2 c+2 a \tan \left (\frac {1}{2} (d+e x)\right )\right )^2-4 \left (a^2-c^2\right )}d\left (2 c+2 a \tan \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2-c^2\right )}}{2 \left (a^2-c^2\right )}+\frac {\left (-2 a^2 C+5 a A c-3 c^2 C\right ) \cos (d+e x)}{2 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^2}}{3 \left (a^2-c^2\right )}+\frac {(A c-a C) \cos (d+e x)}{3 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^3}-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(A c-a C) \cos (d+e x)}{3 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^3}+\frac {\frac {\left (-2 a^2 C+5 a A c-3 c^2 C\right ) \cos (d+e x)}{2 e \left (a^2-c^2\right ) (a+c \sin (d+e x))^2}+\frac {\frac {6 \left (2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (d+e x)\right )+2 c}{2 \sqrt {a^2-c^2}}\right )}{e \left (a^2-c^2\right )^{3/2}}+\frac {\left (-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right ) \cos (d+e x)}{e \left (a^2-c^2\right ) (a+c \sin (d+e x))}}{2 \left (a^2-c^2\right )}}{3 \left (a^2-c^2\right )}-\frac {B}{3 c e (a+c \sin (d+e x))^3}\)

Input:

Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^4,x]
 

Output:

-1/3*B/(c*e*(a + c*Sin[d + e*x])^3) + ((A*c - a*C)*Cos[d + e*x])/(3*(a^2 - 
 c^2)*e*(a + c*Sin[d + e*x])^3) + (((5*a*A*c - 2*a^2*C - 3*c^2*C)*Cos[d + 
e*x])/(2*(a^2 - c^2)*e*(a + c*Sin[d + e*x])^2) + ((6*(2*a^3*A + 3*a*A*c^2 
- 4*a^2*c*C - c^3*C)*ArcTan[(2*c + 2*a*Tan[(d + e*x)/2])/(2*Sqrt[a^2 - c^2 
])])/((a^2 - c^2)^(3/2)*e) + ((11*a^2*A*c + 4*A*c^3 - 2*a^3*C - 13*a*c^2*C 
)*Cos[d + e*x])/((a^2 - c^2)*e*(a + c*Sin[d + e*x])))/(2*(a^2 - c^2)))/(3* 
(a^2 - c^2))
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 25

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

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 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 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*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 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3233

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

rule 4876

Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] : 
> With[{e = FreeFactors[Sin[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + 
Simp[d   Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Sin[ 
c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] &&  !FreeQ[v, x] && Intege 
rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Cos] || EqQ[F, cos])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(714\) vs. \(2(245)=490\).

Time = 1.00 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.77

method	result	size
parts	\(\frac {\frac {\frac {c \left (9 A \,a^{4} c -6 A \,a^{2} c^{3}+2 A \,c^{5}-4 C \,a^{5}-C \,a^{3} c^{2}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{a \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {\left (6 A \,a^{6} c +27 A \,a^{4} c^{3}-12 A \,a^{2} c^{5}+4 A \,c^{7}-2 C \,a^{7}-14 C \,a^{5} c^{2}-11 C \,a^{3} c^{4}+2 C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) a^{2}}+\frac {2 c \left (54 A \,a^{6} c +21 A \,a^{4} c^{3}-4 A \,a^{2} c^{5}+4 A \,c^{7}-18 C \,a^{7}-42 C \,a^{5} c^{2}-17 C \,a^{3} c^{4}+2 C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 a^{3} \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {2 \left (6 A \,a^{6} c +20 A \,a^{4} c^{3}-3 A \,a^{2} c^{5}+2 A \,c^{7}-2 C \,a^{7}-10 C \,a^{5} c^{2}-14 C \,a^{3} c^{4}+C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a^{2} \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {c \left (27 A \,a^{4} c -4 A \,a^{2} c^{3}+2 A \,c^{5}-8 C \,a^{5}-19 C \,a^{3} c^{2}+2 C a \,c^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {2 \left (18 A \,a^{4} c -5 A \,a^{2} c^{3}+2 A \,c^{5}-6 C \,a^{5}-10 C \,a^{3} c^{2}+C a \,c^{4}\right )}{6 a^{6}-18 a^{4} c^{2}+18 a^{2} c^{4}-6 c^{6}}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a +2 c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,c^{2}-4 C \,a^{2} c -C \,c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 c}{2 \sqrt {a^{2}-c^{2}}}\right )}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) \sqrt {a^{2}-c^{2}}}}{e}-\frac {B}{3 c e \left (a +c \sin \left (e x +d \right )\right )^{3}}\)	\(715\)
derivativedivides	\(\frac {\frac {\frac {\left (9 A \,a^{4} c^{2}-6 A \,a^{2} c^{4}+2 A \,c^{6}+2 B \,a^{6}-6 B \,a^{4} c^{2}+6 B \,a^{2} c^{4}-2 B \,c^{6}-4 C \,a^{5} c -C \,a^{3} c^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{a \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {\left (6 A \,a^{6} c +27 A \,a^{4} c^{3}-12 A \,a^{2} c^{5}+4 A \,c^{7}+4 B \,a^{6} c -12 B \,a^{4} c^{3}+12 B \,a^{2} c^{5}-4 B \,c^{7}-2 C \,a^{7}-14 C \,a^{5} c^{2}-11 C \,a^{3} c^{4}+2 C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) a^{2}}+\frac {2 \left (54 A \,a^{6} c^{2}+21 A \,a^{4} c^{4}-4 A \,a^{2} c^{6}+4 A \,c^{8}+6 B \,a^{8}-14 B \,a^{6} c^{2}+6 B \,a^{4} c^{4}+6 B \,a^{2} c^{6}-4 B \,c^{8}-18 C \,a^{7} c -42 C \,a^{5} c^{3}-17 C \,a^{3} c^{5}+2 C a \,c^{7}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 a^{3} \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {2 \left (6 A \,a^{6} c +20 A \,a^{4} c^{3}-3 A \,a^{2} c^{5}+2 A \,c^{7}+2 B \,a^{6} c -6 B \,a^{4} c^{3}+6 B \,a^{2} c^{5}-2 B \,c^{7}-2 C \,a^{7}-10 C \,a^{5} c^{2}-14 C \,a^{3} c^{4}+C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) a^{2}}+\frac {\left (27 A \,a^{4} c^{2}-4 A \,a^{2} c^{4}+2 A \,c^{6}+2 B \,a^{6}-6 B \,a^{4} c^{2}+6 B \,a^{2} c^{4}-2 B \,c^{6}-8 C \,a^{5} c -19 C \,a^{3} c^{3}+2 C a \,c^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {2 \left (18 A \,a^{4} c -5 A \,a^{2} c^{3}+2 A \,c^{5}-6 C \,a^{5}-10 C \,a^{3} c^{2}+C a \,c^{4}\right )}{6 a^{6}-18 a^{4} c^{2}+18 a^{2} c^{4}-6 c^{6}}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a +2 c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,c^{2}-4 C \,a^{2} c -C \,c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 c}{2 \sqrt {a^{2}-c^{2}}}\right )}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) \sqrt {a^{2}-c^{2}}}}{e}\)	\(860\)
default	\(\frac {\frac {\frac {\left (9 A \,a^{4} c^{2}-6 A \,a^{2} c^{4}+2 A \,c^{6}+2 B \,a^{6}-6 B \,a^{4} c^{2}+6 B \,a^{2} c^{4}-2 B \,c^{6}-4 C \,a^{5} c -C \,a^{3} c^{3}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{a \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {\left (6 A \,a^{6} c +27 A \,a^{4} c^{3}-12 A \,a^{2} c^{5}+4 A \,c^{7}+4 B \,a^{6} c -12 B \,a^{4} c^{3}+12 B \,a^{2} c^{5}-4 B \,c^{7}-2 C \,a^{7}-14 C \,a^{5} c^{2}-11 C \,a^{3} c^{4}+2 C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) a^{2}}+\frac {2 \left (54 A \,a^{6} c^{2}+21 A \,a^{4} c^{4}-4 A \,a^{2} c^{6}+4 A \,c^{8}+6 B \,a^{8}-14 B \,a^{6} c^{2}+6 B \,a^{4} c^{4}+6 B \,a^{2} c^{6}-4 B \,c^{8}-18 C \,a^{7} c -42 C \,a^{5} c^{3}-17 C \,a^{3} c^{5}+2 C a \,c^{7}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 a^{3} \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {2 \left (6 A \,a^{6} c +20 A \,a^{4} c^{3}-3 A \,a^{2} c^{5}+2 A \,c^{7}+2 B \,a^{6} c -6 B \,a^{4} c^{3}+6 B \,a^{2} c^{5}-2 B \,c^{7}-2 C \,a^{7}-10 C \,a^{5} c^{2}-14 C \,a^{3} c^{4}+C a \,c^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) a^{2}}+\frac {\left (27 A \,a^{4} c^{2}-4 A \,a^{2} c^{4}+2 A \,c^{6}+2 B \,a^{6}-6 B \,a^{4} c^{2}+6 B \,a^{2} c^{4}-2 B \,c^{6}-8 C \,a^{5} c -19 C \,a^{3} c^{3}+2 C a \,c^{5}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right )}+\frac {2 \left (18 A \,a^{4} c -5 A \,a^{2} c^{3}+2 A \,c^{5}-6 C \,a^{5}-10 C \,a^{3} c^{2}+C a \,c^{4}\right )}{6 a^{6}-18 a^{4} c^{2}+18 a^{2} c^{4}-6 c^{6}}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} a +2 c \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+a \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,c^{2}-4 C \,a^{2} c -C \,c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 c}{2 \sqrt {a^{2}-c^{2}}}\right )}{\left (a^{6}-3 a^{4} c^{2}+3 a^{2} c^{4}-c^{6}\right ) \sqrt {a^{2}-c^{2}}}}{e}\)	\(860\)
risch	\(\text {Expression too large to display}\)	\(1249\)

Input:

int((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+c*sin(e*x+d))^4,x,method=_RETURNVERBO 
SE)
 

Output:

1/e*(2*(1/2*c*(9*A*a^4*c-6*A*a^2*c^3+2*A*c^5-4*C*a^5-C*a^3*c^2)/a/(a^6-3*a 
^4*c^2+3*a^2*c^4-c^6)*tan(1/2*e*x+1/2*d)^5+1/2*(6*A*a^6*c+27*A*a^4*c^3-12* 
A*a^2*c^5+4*A*c^7-2*C*a^7-14*C*a^5*c^2-11*C*a^3*c^4+2*C*a*c^6)/(a^6-3*a^4* 
c^2+3*a^2*c^4-c^6)/a^2*tan(1/2*e*x+1/2*d)^4+1/3/a^3*c*(54*A*a^6*c+21*A*a^4 
*c^3-4*A*a^2*c^5+4*A*c^7-18*C*a^7-42*C*a^5*c^2-17*C*a^3*c^4+2*C*a*c^6)/(a^ 
6-3*a^4*c^2+3*a^2*c^4-c^6)*tan(1/2*e*x+1/2*d)^3+1/a^2*(6*A*a^6*c+20*A*a^4* 
c^3-3*A*a^2*c^5+2*A*c^7-2*C*a^7-10*C*a^5*c^2-14*C*a^3*c^4+C*a*c^6)/(a^6-3* 
a^4*c^2+3*a^2*c^4-c^6)*tan(1/2*e*x+1/2*d)^2+1/2*c*(27*A*a^4*c-4*A*a^2*c^3+ 
2*A*c^5-8*C*a^5-19*C*a^3*c^2+2*C*a*c^4)/a/(a^6-3*a^4*c^2+3*a^2*c^4-c^6)*ta 
n(1/2*e*x+1/2*d)+1/6*(18*A*a^4*c-5*A*a^2*c^3+2*A*c^5-6*C*a^5-10*C*a^3*c^2+ 
C*a*c^4)/(a^6-3*a^4*c^2+3*a^2*c^4-c^6))/(tan(1/2*e*x+1/2*d)^2*a+2*c*tan(1/ 
2*e*x+1/2*d)+a)^3+(2*A*a^3+3*A*a*c^2-4*C*a^2*c-C*c^3)/(a^6-3*a^4*c^2+3*a^2 
*c^4-c^6)/(a^2-c^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*e*x+1/2*d)+2*c)/(a^2-c^2 
)^(1/2)))-1/3*B/c/e/(a+c*sin(e*x+d))^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 664 vs. \(2 (245) = 490\).

Time = 0.15 (sec) , antiderivative size = 1411, normalized size of antiderivative = 5.47 \[ \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+c*sin(e*x+d))^4,x, algorithm="f 
ricas")
 

Output:

[1/12*(4*B*a^8 - 16*B*a^6*c^2 + 24*B*a^4*c^4 - 16*B*a^2*c^6 + 4*B*c^8 - 2* 
(2*C*a^5*c^3 - 11*A*a^4*c^4 + 11*C*a^3*c^5 + 7*A*a^2*c^6 - 13*C*a*c^7 + 4* 
A*c^8)*cos(e*x + d)^3 + 6*(2*C*a^6*c^2 - 9*A*a^5*c^3 + 7*C*a^4*c^4 + 8*A*a 
^3*c^5 - 10*C*a^2*c^6 + A*a*c^7 + C*c^8)*cos(e*x + d)*sin(e*x + d) + 3*(2* 
A*a^6*c - 4*C*a^5*c^2 + 9*A*a^4*c^3 - 13*C*a^3*c^4 + 9*A*a^2*c^5 - 3*C*a*c 
^6 - 3*(2*A*a^4*c^3 - 4*C*a^3*c^4 + 3*A*a^2*c^5 - C*a*c^6)*cos(e*x + d)^2 
+ (6*A*a^5*c^2 - 12*C*a^4*c^3 + 11*A*a^3*c^4 - 7*C*a^2*c^5 + 3*A*a*c^6 - C 
*c^7 - (2*A*a^3*c^4 - 4*C*a^2*c^5 + 3*A*a*c^6 - C*c^7)*cos(e*x + d)^2)*sin 
(e*x + d))*sqrt(-a^2 + c^2)*log(((2*a^2 - c^2)*cos(e*x + d)^2 - 2*a*c*sin( 
e*x + d) - a^2 - c^2 + 2*(a*cos(e*x + d)*sin(e*x + d) + c*cos(e*x + d))*sq 
rt(-a^2 + c^2))/(c^2*cos(e*x + d)^2 - 2*a*c*sin(e*x + d) - a^2 - c^2)) + 1 
2*(C*a^7*c - 3*A*a^6*c^2 + C*a^5*c^3 + 2*A*a^4*c^4 - 2*C*a*c^7 + A*c^8)*co 
s(e*x + d))/(3*(a^9*c^3 - 4*a^7*c^5 + 6*a^5*c^7 - 4*a^3*c^9 + a*c^11)*e*co 
s(e*x + d)^2 - (a^11*c - a^9*c^3 - 6*a^7*c^5 + 14*a^5*c^7 - 11*a^3*c^9 + 3 
*a*c^11)*e + ((a^8*c^4 - 4*a^6*c^6 + 6*a^4*c^8 - 4*a^2*c^10 + c^12)*e*cos( 
e*x + d)^2 - (3*a^10*c^2 - 11*a^8*c^4 + 14*a^6*c^6 - 6*a^4*c^8 - a^2*c^10 
+ c^12)*e)*sin(e*x + d)), 1/6*(2*B*a^8 - 8*B*a^6*c^2 + 12*B*a^4*c^4 - 8*B* 
a^2*c^6 + 2*B*c^8 - (2*C*a^5*c^3 - 11*A*a^4*c^4 + 11*C*a^3*c^5 + 7*A*a^2*c 
^6 - 13*C*a*c^7 + 4*A*c^8)*cos(e*x + d)^3 + 3*(2*C*a^6*c^2 - 9*A*a^5*c^3 + 
 7*C*a^4*c^4 + 8*A*a^3*c^5 - 10*C*a^2*c^6 + A*a*c^7 + C*c^8)*cos(e*x + ...
 

Sympy [F(-1)]

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

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+c*sin(e*x+d))**4,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+c*sin(e*x+d))^4,x, algorithm="m 
axima")
 

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*c^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. 1281 vs. \(2 (245) = 490\).

Time = 0.36 (sec) , antiderivative size = 1281, normalized size of antiderivative = 4.97 \[ \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+c*sin(e*x+d))^4,x, algorithm="g 
iac")
 

Output:

1/3*(3*(2*A*a^3 - 4*C*a^2*c + 3*A*a*c^2 - C*c^3)*(pi*floor(1/2*(e*x + d)/p 
i + 1/2)*sgn(a) + arctan((a*tan(1/2*e*x + 1/2*d) + c)/sqrt(a^2 - c^2)))/(( 
a^6 - 3*a^4*c^2 + 3*a^2*c^4 - c^6)*sqrt(a^2 - c^2)) + (6*B*a^8*tan(1/2*e*x 
 + 1/2*d)^5 - 12*C*a^7*c*tan(1/2*e*x + 1/2*d)^5 + 27*A*a^6*c^2*tan(1/2*e*x 
 + 1/2*d)^5 - 18*B*a^6*c^2*tan(1/2*e*x + 1/2*d)^5 - 3*C*a^5*c^3*tan(1/2*e* 
x + 1/2*d)^5 - 18*A*a^4*c^4*tan(1/2*e*x + 1/2*d)^5 + 18*B*a^4*c^4*tan(1/2* 
e*x + 1/2*d)^5 + 6*A*a^2*c^6*tan(1/2*e*x + 1/2*d)^5 - 6*B*a^2*c^6*tan(1/2* 
e*x + 1/2*d)^5 - 6*C*a^8*tan(1/2*e*x + 1/2*d)^4 + 18*A*a^7*c*tan(1/2*e*x + 
 1/2*d)^4 + 12*B*a^7*c*tan(1/2*e*x + 1/2*d)^4 - 42*C*a^6*c^2*tan(1/2*e*x + 
 1/2*d)^4 + 81*A*a^5*c^3*tan(1/2*e*x + 1/2*d)^4 - 36*B*a^5*c^3*tan(1/2*e*x 
 + 1/2*d)^4 - 33*C*a^4*c^4*tan(1/2*e*x + 1/2*d)^4 - 36*A*a^3*c^5*tan(1/2*e 
*x + 1/2*d)^4 + 36*B*a^3*c^5*tan(1/2*e*x + 1/2*d)^4 + 6*C*a^2*c^6*tan(1/2* 
e*x + 1/2*d)^4 + 12*A*a*c^7*tan(1/2*e*x + 1/2*d)^4 - 12*B*a*c^7*tan(1/2*e* 
x + 1/2*d)^4 + 12*B*a^8*tan(1/2*e*x + 1/2*d)^3 - 36*C*a^7*c*tan(1/2*e*x + 
1/2*d)^3 + 108*A*a^6*c^2*tan(1/2*e*x + 1/2*d)^3 - 28*B*a^6*c^2*tan(1/2*e*x 
 + 1/2*d)^3 - 84*C*a^5*c^3*tan(1/2*e*x + 1/2*d)^3 + 42*A*a^4*c^4*tan(1/2*e 
*x + 1/2*d)^3 + 12*B*a^4*c^4*tan(1/2*e*x + 1/2*d)^3 - 34*C*a^3*c^5*tan(1/2 
*e*x + 1/2*d)^3 - 8*A*a^2*c^6*tan(1/2*e*x + 1/2*d)^3 + 12*B*a^2*c^6*tan(1/ 
2*e*x + 1/2*d)^3 + 4*C*a*c^7*tan(1/2*e*x + 1/2*d)^3 + 8*A*c^8*tan(1/2*e*x 
+ 1/2*d)^3 - 8*B*c^8*tan(1/2*e*x + 1/2*d)^3 - 12*C*a^8*tan(1/2*e*x + 1/...
 

Mupad [B] (verification not implemented)

Time = 18.86 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.21 \[ \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx =\text {Too large to display} \] Input:

int((A + B*cos(d + e*x) + C*sin(d + e*x))/(a + c*sin(d + e*x))^4,x)
 

Output:

((2*A*c^5 - 6*C*a^5 - 5*A*a^2*c^3 - 10*C*a^3*c^2 + 18*A*a^4*c + C*a*c^4)/( 
3*(a^6 - c^6 + 3*a^2*c^4 - 3*a^4*c^2)) + (tan(d/2 + (e*x)/2)*(2*B*a^6 + 2* 
A*c^6 - 2*B*c^6 - 4*A*a^2*c^4 + 27*A*a^4*c^2 + 6*B*a^2*c^4 - 6*B*a^4*c^2 - 
 19*C*a^3*c^3 + 2*C*a*c^5 - 8*C*a^5*c))/(a*(a^6 - c^6 + 3*a^2*c^4 - 3*a^4* 
c^2)) + (2*tan(d/2 + (e*x)/2)^2*(2*A*c^7 - 2*C*a^7 - 2*B*c^7 - 3*A*a^2*c^5 
 + 20*A*a^4*c^3 + 6*B*a^2*c^5 - 6*B*a^4*c^3 - 14*C*a^3*c^4 - 10*C*a^5*c^2 
+ 6*A*a^6*c + 2*B*a^6*c + C*a*c^6))/(a^2*(a^6 - c^6 + 3*a^2*c^4 - 3*a^4*c^ 
2)) + (tan(d/2 + (e*x)/2)^4*(4*A*c^7 - 2*C*a^7 - 4*B*c^7 - 12*A*a^2*c^5 + 
27*A*a^4*c^3 + 12*B*a^2*c^5 - 12*B*a^4*c^3 - 11*C*a^3*c^4 - 14*C*a^5*c^2 + 
 6*A*a^6*c + 4*B*a^6*c + 2*C*a*c^6))/(a^2*(a^6 - c^6 + 3*a^2*c^4 - 3*a^4*c 
^2)) - (tan(d/2 + (e*x)/2)^5*(2*B*c^6 - 2*A*c^6 - 2*B*a^6 + 6*A*a^2*c^4 - 
9*A*a^4*c^2 - 6*B*a^2*c^4 + 6*B*a^4*c^2 + C*a^3*c^3 + 4*C*a^5*c))/(a*(a^6 
- c^6 + 3*a^2*c^4 - 3*a^4*c^2)) + (2*tan(d/2 + (e*x)/2)^3*(3*a^2 + 2*c^2)* 
(2*B*a^6 + 2*A*c^6 - 2*B*c^6 - 5*A*a^2*c^4 + 18*A*a^4*c^2 + 6*B*a^2*c^4 - 
6*B*a^4*c^2 - 10*C*a^3*c^3 + C*a*c^5 - 6*C*a^5*c))/(3*a^3*(a^6 - c^6 + 3*a 
^2*c^4 - 3*a^4*c^2)))/(e*(a^3*tan(d/2 + (e*x)/2)^6 + tan(d/2 + (e*x)/2)^2* 
(12*a*c^2 + 3*a^3) + tan(d/2 + (e*x)/2)^4*(12*a*c^2 + 3*a^3) + tan(d/2 + ( 
e*x)/2)^3*(12*a^2*c + 8*c^3) + a^3 + 6*a^2*c*tan(d/2 + (e*x)/2) + 6*a^2*c* 
tan(d/2 + (e*x)/2)^5)) + (atan(((((2*A*a^3 - C*c^3 + 3*A*a*c^2 - 4*C*a^2*c 
)*(2*a^6*c - 2*c^7 + 6*a^2*c^5 - 6*a^4*c^3))/(2*(a + c)^(7/2)*(a - c)^(...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 954, normalized size of antiderivative = 3.70 \[ \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx =\text {Too large to display} \] Input:

int((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+c*sin(e*x+d))^4,x)
 

Output:

(12*sqrt(a**2 - c**2)*atan((tan((d + e*x)/2)*a + c)/sqrt(a**2 - c**2))*sin 
(d + e*x)**3*a**4*c**4 + 6*sqrt(a**2 - c**2)*atan((tan((d + e*x)/2)*a + c) 
/sqrt(a**2 - c**2))*sin(d + e*x)**3*a**2*c**6 + 36*sqrt(a**2 - c**2)*atan( 
(tan((d + e*x)/2)*a + c)/sqrt(a**2 - c**2))*sin(d + e*x)**2*a**5*c**3 + 18 
*sqrt(a**2 - c**2)*atan((tan((d + e*x)/2)*a + c)/sqrt(a**2 - c**2))*sin(d 
+ e*x)**2*a**3*c**5 + 36*sqrt(a**2 - c**2)*atan((tan((d + e*x)/2)*a + c)/s 
qrt(a**2 - c**2))*sin(d + e*x)*a**6*c**2 + 18*sqrt(a**2 - c**2)*atan((tan( 
(d + e*x)/2)*a + c)/sqrt(a**2 - c**2))*sin(d + e*x)*a**4*c**4 + 12*sqrt(a* 
*2 - c**2)*atan((tan((d + e*x)/2)*a + c)/sqrt(a**2 - c**2))*a**7*c + 6*sqr 
t(a**2 - c**2)*atan((tan((d + e*x)/2)*a + c)/sqrt(a**2 - c**2))*a**5*c**3 
+ 9*cos(d + e*x)*sin(d + e*x)**2*a**5*c**4 - 9*cos(d + e*x)*sin(d + e*x)** 
2*a**3*c**6 + 21*cos(d + e*x)*sin(d + e*x)*a**6*c**3 - 24*cos(d + e*x)*sin 
(d + e*x)*a**4*c**5 + 3*cos(d + e*x)*sin(d + e*x)*a**2*c**7 + 12*cos(d + e 
*x)*a**7*c**2 - 15*cos(d + e*x)*a**5*c**4 + 3*cos(d + e*x)*a**3*c**6 + 7*s 
in(d + e*x)**3*a**4*c**5 - 8*sin(d + e*x)**3*a**2*c**7 + sin(d + e*x)**3*c 
**9 + 21*sin(d + e*x)**2*a**5*c**4 - 24*sin(d + e*x)**2*a**3*c**6 + 3*sin( 
d + e*x)**2*a*c**8 + 21*sin(d + e*x)*a**6*c**3 - 24*sin(d + e*x)*a**4*c**5 
 + 3*sin(d + e*x)*a**2*c**7 - 2*a**8*b + 7*a**7*c**2 + 6*a**6*b*c**2 - 8*a 
**5*c**4 - 6*a**4*b*c**4 + a**3*c**6 + 2*a**2*b*c**6)/(6*a**2*c*e*(sin(d + 
 e*x)**3*a**6*c**3 - 3*sin(d + e*x)**3*a**4*c**5 + 3*sin(d + e*x)**3*a*...