\(\int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx\) [469]

Optimal result

Integrand size = 21, antiderivative size = 300 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (4 a^2-7 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:

1/2*(12*a^2+b^2)*x/b^5-a^3*(12*a^4-29*a^2*b^2+20*b^4)*arctan((a-b)^(1/2)*t 
an(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^5/(a+b)^(5/2)/d-3/2*a*(4*a^4- 
7*a^2*b^2+2*b^4)*sin(d*x+c)/b^4/(a^2-b^2)^2/d+1/2*(6*a^4-10*a^2*b^2+b^4)*c 
os(d*x+c)*sin(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*a^2*cos(d*x+c)^3*sin(d*x+c)/b/( 
a^2-b^2)/d/(a+b*cos(d*x+c))^2-1/2*a^2*(4*a^2-7*b^2)*cos(d*x+c)^2*sin(d*x+c 
)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
 

Mathematica [A] (verified)

Time = 2.64 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.66 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {2 \left (12 a^2+b^2\right ) (c+d x)+\frac {4 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}-12 a b \sin (c+d x)+\frac {2 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {2 a^4 b \left (-7 a^2+10 b^2\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+b^2 \sin (2 (c+d x))}{4 b^5 d} \] Input:

Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3,x]
 

Output:

(2*(12*a^2 + b^2)*(c + d*x) + (4*a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan 
h[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 12*a* 
b*Sin[c + d*x] + (2*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d* 
x])^2) + (2*a^4*b*(-7*a^2 + 10*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a 
+ b*Cos[c + d*x])) + b^2*Sin[2*(c + d*x)])/(4*b^5*d)
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3271, 3042, 3526, 25, 3042, 3528, 27, 3042, 3502, 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.

\(\displaystyle \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3271

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int -\frac {\cos (c+d x) \left (2 \left (4 a^2-7 b^2\right ) a^2-b \left (a^2-4 b^2\right ) \cos (c+d x) a-2 \left (6 a^4-10 b^2 a^2+b^4\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {\cos (c+d x) \left (2 \left (4 a^2-7 b^2\right ) a^2-b \left (a^2-4 b^2\right ) \cos (c+d x) a-2 \left (6 a^4-10 b^2 a^2+b^4\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 \left (4 a^2-7 b^2\right ) a^2-b \left (a^2-4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-2 \left (6 a^4-10 b^2 a^2+b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {\frac {\int -\frac {2 \left (-3 a \left (4 a^4-7 b^2 a^2+2 b^4\right ) \cos ^2(c+d x)-b \left (2 a^4-4 b^2 a^2-b^4\right ) \cos (c+d x)+a \left (6 a^4-10 b^2 a^2+b^4\right )\right )}{a+b \cos (c+d x)}dx}{2 b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {-\frac {\int \frac {-3 a \left (4 a^4-7 b^2 a^2+2 b^4\right ) \cos ^2(c+d x)-b \left (2 a^4-4 b^2 a^2-b^4\right ) \cos (c+d x)+a \left (6 a^4-10 b^2 a^2+b^4\right )}{a+b \cos (c+d x)}dx}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\int \frac {-3 a \left (4 a^4-7 b^2 a^2+2 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (2 a^4-4 b^2 a^2-b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (6 a^4-10 b^2 a^2+b^4\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3502

\(\displaystyle -\frac {\frac {-\frac {\frac {\int \frac {\left (12 a^2+b^2\right ) \cos (c+d x) \left (a^2-b^2\right )^2+a b \left (6 a^4-10 b^2 a^2+b^4\right )}{a+b \cos (c+d x)}dx}{b}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\frac {\int \frac {\left (12 a^2+b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2+a b \left (6 a^4-10 b^2 a^2+b^4\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3214

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right )}{b}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}}{b}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right )}{b}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3138

\(\displaystyle -\frac {\frac {-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right )}{b}-\frac {2 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {a^2 \left (4 a^2-7 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right )}{b}-\frac {2 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\)

Input:

Int[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3,x]
 

Output:

-1/2*(a^2*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x 
])^2) - ((a^2*(4*a^2 - 7*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)* 
d*(a + b*Cos[c + d*x])) + (-(((6*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[ 
c + d*x])/(b*d)) - ((((a^2 - b^2)^2*(12*a^2 + b^2)*x)/b - (2*a^3*(12*a^4 - 
 29*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/ 
(Sqrt[a - b]*b*Sqrt[a + b]*d))/b - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c 
+ d*x])/(b*d))/b)/(b*(a^2 - b^2)))/(2*b*(a^2 - b^2))
 

Defintions of rubi rules used

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 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 3271

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co 
s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* 
(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin 
[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 
2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 
+ b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - 
 d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] 
, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - 
b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || 
IntegersQ[2*m, 2*n])
 

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 3526

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^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d 
, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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 = 2.15 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-3 a b -\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-3 a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (12 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {2 a^{3} \left (\frac {\frac {\left (6 a^{2}-a b -10 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (6 a^{2}+a b -10 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (12 a^{4}-29 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(306\)
default	\(\frac {\frac {\frac {2 \left (\left (-3 a b -\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-3 a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (12 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}-\frac {2 a^{3} \left (\frac {\frac {\left (6 a^{2}-a b -10 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (6 a^{2}+a b -10 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (12 a^{4}-29 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(306\)
risch	\(\frac {6 x \,a^{2}}{b^{5}}+\frac {x}{2 b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,b^{3}}+\frac {3 i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,b^{4}}-\frac {3 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,b^{4}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,b^{3}}-\frac {i a^{4} \left (8 a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}-11 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-13 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-10 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+20 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-29 b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}+7 a^{2} b^{2}-10 b^{4}\right )}{b^{5} \left (a^{2}-b^{2}\right )^{2} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )^{2}}-\frac {6 a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{5}}+\frac {29 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}-\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}+\frac {6 a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{5}}-\frac {29 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}+\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}\)	\(782\)

Input:

int(cos(d*x+c)^5/(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/d*(2/b^5*(((-3*a*b-1/2*b^2)*tan(1/2*d*x+1/2*c)^3+(-3*a*b+1/2*b^2)*tan(1/ 
2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(12*a^2+b^2)*arctan(tan(1/2*d 
*x+1/2*c)))-2*a^3/b^5*((1/2*(6*a^2-a*b-10*b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*t 
an(1/2*d*x+1/2*c)^3+1/2*(6*a^2+a*b-10*b^2)*a*b/(a+b)/(a^2-2*a*b+b^2)*tan(1 
/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^2+1/2*( 
12*a^4-29*a^2*b^2+20*b^4)/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(( 
a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1161, normalized size of antiderivative = 3.87 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
 

Output:

[1/4*(2*(12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d*x*cos( 
d*x + c)^2 + 4*(12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d* 
x*cos(d*x + c) + 2*(12*a^10 - 35*a^8*b^2 + 33*a^6*b^4 - 9*a^4*b^6 - a^2*b^ 
8)*d*x - (12*a^9 - 29*a^7*b^2 + 20*a^5*b^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20 
*a^3*b^6)*cos(d*x + c)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*cos(d*x 
+ c))*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)) - 2*(12*a^9*b - 33*a^7*b^ 
3 + 27*a^5*b^5 - 6*a^3*b^7 - (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos( 
d*x + c)^3 + 4*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(d*x + c)^2 + 
(18*a^8*b^2 - 50*a^6*b^4 + 43*a^4*b^6 - 11*a^2*b^8)*cos(d*x + c))*sin(d*x 
+ c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2 + 2*(a^7 
*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c) + (a^8*b^5 - 3*a^6* 
b^7 + 3*a^4*b^9 - a^2*b^11)*d), 1/2*((12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 
 - 9*a^2*b^8 - b^10)*d*x*cos(d*x + c)^2 + 2*(12*a^9*b - 35*a^7*b^3 + 33*a^ 
5*b^5 - 9*a^3*b^7 - a*b^9)*d*x*cos(d*x + c) + (12*a^10 - 35*a^8*b^2 + 33*a 
^6*b^4 - 9*a^4*b^6 - a^2*b^8)*d*x - (12*a^9 - 29*a^7*b^2 + 20*a^5*b^4 + (1 
2*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(d*x + c)^2 + 2*(12*a^8*b - 29*a^6 
*b^3 + 20*a^4*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + 
 b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (12*a^9*b - 33*a^7*b^3 + 27*a^5*b...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**5/(a+b*cos(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,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. 1735 vs. \(2 (281) = 562\).

Time = 0.66 (sec) , antiderivative size = 1735, normalized size of antiderivative = 5.78 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,x, algorithm="giac")
 

Output:

-1/2*(((12*a^6 - 6*a^5*b - 23*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 - a*b^5 + 
b^6)*sqrt(a^2 - b^2)*abs(a^4*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) + (24*a^11 
*b^4 - 12*a^10*b^5 - 100*a^9*b^6 + 47*a^8*b^7 + 158*a^7*b^8 - 68*a^6*b^9 - 
 111*a^5*b^10 + 42*a^4*b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14 - b^15)*sq 
rt(a^2 - b^2)*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*ta 
n(1/2*d*x + 1/2*c)/sqrt((4*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 + sqrt(-16*(a^5*b 
^4 + a^4*b^5 - 2*a^3*b^6 - 2*a^2*b^7 + a*b^8 + b^9)*(a^5*b^4 - a^4*b^5 - 2 
*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9) + 16*(a^5*b^4 - 2*a^3*b^6 + a*b^8)^2)) 
/(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9))))/((a^4*b^5 - 
2*a^2*b^7 + b^9)^2*(a^2 - 2*a*b + b^2) + (a^7*b^4 - 2*a^6*b^5 - a^5*b^6 + 
4*a^4*b^7 - a^3*b^8 - 2*a^2*b^9 + a*b^10)*abs(a^4*b^5 - 2*a^2*b^7 + b^9)) 
- (24*a^11*b^4 - 12*a^10*b^5 - 100*a^9*b^6 + 47*a^8*b^7 + 158*a^7*b^8 - 68 
*a^6*b^9 - 111*a^5*b^10 + 42*a^4*b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14 
- b^15 - 12*a^6*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + 6*a^5*b*abs(a^4*b^5 - 2*a 
^2*b^7 + b^9) + 23*a^4*b^2*abs(a^4*b^5 - 2*a^2*b^7 + b^9) - 10*a^3*b^3*abs 
(a^4*b^5 - 2*a^2*b^7 + b^9) - 10*a^2*b^4*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + 
a*b^5*abs(a^4*b^5 - 2*a^2*b^7 + b^9) - b^6*abs(a^4*b^5 - 2*a^2*b^7 + b^9)) 
*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/2*c)/sqrt((4 
*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 - sqrt(-16*(a^5*b^4 + a^4*b^5 - 2*a^3*b^6 - 
 2*a^2*b^7 + a*b^8 + b^9)*(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + ...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 50.59 (sec) , antiderivative size = 5962, normalized size of antiderivative = 19.87 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^5/(a + b*cos(c + d*x))^3,x)
 

Output:

(atan(((((8*tan(c/2 + (d*x)/2)*(288*a^14 - 288*a^13*b - 2*a*b^13 + b^14 + 
21*a^2*b^12 - 40*a^3*b^11 + 74*a^4*b^10 - 108*a^5*b^9 + 18*a^6*b^8 + 872*a 
^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 1538*a^10*b^4 + 1104*a^11*b^3 - 1104 
*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5* 
b^10 - a^6*b^9 - a^7*b^8) + (((4*(4*b^21 + 28*a^2*b^19 - 80*a^3*b^18 - 120 
*a^4*b^17 + 276*a^5*b^16 + 164*a^6*b^15 - 360*a^7*b^14 - 100*a^8*b^13 + 21 
2*a^9*b^12 + 24*a^10*b^11 - 48*a^11*b^10))/(a*b^18 + b^19 - 3*a^2*b^17 - 3 
*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - (4*tan(c/2 + 
(d*x)/2)*(a^2*12i + b^2*1i)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4* 
b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 
- 8*a^10*b^10))/(b^5*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 
 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(a^2*12i + b^2*1i))/(2*b^5))*(a^2*12i 
 + b^2*1i)*1i)/(2*b^5) + (((8*tan(c/2 + (d*x)/2)*(288*a^14 - 288*a^13*b - 
2*a*b^13 + b^14 + 21*a^2*b^12 - 40*a^3*b^11 + 74*a^4*b^10 - 108*a^5*b^9 + 
18*a^6*b^8 + 872*a^7*b^7 - 827*a^8*b^6 - 1538*a^9*b^5 + 1538*a^10*b^4 + 11 
04*a^11*b^3 - 1104*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3 
*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) - (((4*(4*b^21 + 28*a^2*b^19 - 
 80*a^3*b^18 - 120*a^4*b^17 + 276*a^5*b^16 + 164*a^6*b^15 - 360*a^7*b^14 - 
 100*a^8*b^13 + 212*a^9*b^12 + 24*a^10*b^11 - 48*a^11*b^10))/(a*b^18 + b^1 
9 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1470, normalized size of antiderivative = 4.90 \[ \int \frac {\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx =\text {Too large to display} \] Input:

int(cos(d*x+c)^5/(a+b*cos(d*x+c))^3,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))*cos(c + d*x)*a**8*b + 116*sqrt(a**2 - b**2)*atan((tan((c 
+ d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**6*b** 
3 - 80*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq 
rt(a**2 - b**2))*cos(c + d*x)*a**4*b**5 + 24*sqrt(a**2 - b**2)*atan((tan(( 
c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a** 
7*b**2 - 58*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)* 
b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**5*b**4 + 40*sqrt(a**2 - b**2)*ata 
n((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*sin(c + d*x 
)**2*a**3*b**6 - 24*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + 
d*x)/2)*b)/sqrt(a**2 - b**2))*a**9 + 34*sqrt(a**2 - b**2)*atan((tan((c + d 
*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*a**7*b**2 + 18*sqrt(a**2 
 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2)) 
*a**5*b**4 - 40*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x) 
/2)*b)/sqrt(a**2 - b**2))*a**3*b**6 - cos(c + d*x)*sin(c + d*x)**3*a**6*b* 
*4 + 3*cos(c + d*x)*sin(c + d*x)**3*a**4*b**6 - 3*cos(c + d*x)*sin(c + d*x 
)**3*a**2*b**8 + cos(c + d*x)*sin(c + d*x)**3*b**10 - 18*cos(c + d*x)*sin( 
c + d*x)*a**8*b**2 + 51*cos(c + d*x)*sin(c + d*x)*a**6*b**4 - 46*cos(c + d 
*x)*sin(c + d*x)*a**4*b**6 + 14*cos(c + d*x)*sin(c + d*x)*a**2*b**8 - cos( 
c + d*x)*sin(c + d*x)*b**10 + 24*cos(c + d*x)*a**9*b*c + 24*cos(c + d*x...