\(\int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1081]

Optimal result

Integrand size = 29, antiderivative size = 266 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{3/2} d}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \] Output:

-1/2*(12*a^2-b^2)*x/b^5+a*(12*a^4-19*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*d* 
x+1/2*c))/(a^2-b^2)^(1/2))/b^5/(a^2-b^2)^(3/2)/d-1/2*a*(12*a^2-11*b^2)*cos 
(d*x+c)/b^4/(a^2-b^2)/d+1/2*(6*a^2-5*b^2)*cos(d*x+c)*sin(d*x+c)/b^3/(a^2-b 
^2)/d-1/2*cos(d*x+c)*sin(d*x+c)^3/b/d/(a+b*sin(d*x+c))^2-1/2*(4*a^2-3*b^2) 
*cos(d*x+c)*sin(d*x+c)^2/b^2/(a^2-b^2)/d/(a+b*sin(d*x+c))
 

Mathematica [A] (verified)

Time = 7.37 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {4 a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {-4 a b \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \sin (c+d x)-2 b^2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \sin ^2(c+d x)+\cos (c+d x) \left (-24 a^5 b+22 a^3 b^3-8 a b^3 \left (a^2-b^2\right ) \sin ^2(c+d x)+2 b^4 \left (a^2-b^2\right ) \sin ^3(c+d x)\right )-a^2 \left (2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x)+\left (18 a^2 b^2-17 b^4\right ) \sin (2 (c+d x))\right )}{(a+b \sin (c+d x))^2}}{4 (a-b) b^5 (a+b) d} \] Input:

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

Output:

((4*a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a 
^2 - b^2]])/Sqrt[a^2 - b^2] + (-4*a*b*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x 
)*Sin[c + d*x] - 2*b^2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Sin[c + d*x]^ 
2 + Cos[c + d*x]*(-24*a^5*b + 22*a^3*b^3 - 8*a*b^3*(a^2 - b^2)*Sin[c + d*x 
]^2 + 2*b^4*(a^2 - b^2)*Sin[c + d*x]^3) - a^2*(2*(12*a^4 - 13*a^2*b^2 + b^ 
4)*(c + d*x) + (18*a^2*b^2 - 17*b^4)*Sin[2*(c + d*x)]))/(a + b*Sin[c + d*x 
])^2)/(4*(a - b)*b^5*(a + b)*d)
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.17, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 3368, 3042, 3527, 25, 3042, 3527, 25, 3042, 3528, 27, 3042, 3502, 3042, 3214, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3527

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3527

\(\displaystyle \frac {-\frac {\int -\frac {\sin (c+d x) \left (-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (4 a^4-7 b^2 a^2+3 b^4\right )\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\sin (c+d x) \left (-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (4 a^4-7 b^2 a^2+3 b^4\right )\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sin (c+d x) \left (-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)^2-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (4 a^4-7 b^2 a^2+3 b^4\right )\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\frac {\int -\frac {2 \left (-a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)-b \left (2 a^4-3 b^2 a^2+b^4\right ) \sin (c+d x)+a \left (6 a^4-11 b^2 a^2+5 b^4\right )\right )}{a+b \sin (c+d x)}dx}{2 b}+\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\int \frac {-a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)-b \left (2 a^4-3 b^2 a^2+b^4\right ) \sin (c+d x)+a \left (6 a^4-11 b^2 a^2+5 b^4\right )}{a+b \sin (c+d x)}dx}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\int \frac {-a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)^2-b \left (2 a^4-3 b^2 a^2+b^4\right ) \sin (c+d x)+a \left (6 a^4-11 b^2 a^2+5 b^4\right )}{a+b \sin (c+d x)}dx}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\int \frac {\left (12 a^2-b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2+a b \left (6 a^4-11 b^2 a^2+5 b^4\right )}{a+b \sin (c+d x)}dx}{b}+\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\int \frac {\left (12 a^2-b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2+a b \left (6 a^4-11 b^2 a^2+5 b^4\right )}{a+b \sin (c+d x)}dx}{b}+\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right )}{b}-\frac {a \left (12 a^6-31 a^4 b^2+25 a^2 b^4-6 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{b}+\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right )}{b}-\frac {a \left (12 a^6-31 a^4 b^2+25 a^2 b^4-6 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}}{b}+\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right )}{b}-\frac {2 a \left (12 a^6-31 a^4 b^2+25 a^2 b^4-6 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b}+\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\frac {4 a \left (12 a^6-31 a^4 b^2+25 a^2 b^4-6 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {x \left (12 a^2-b^2\right ) \left (a^2-b^2\right )^2}{b}}{b}+\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {\left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b d}+\frac {\frac {x \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right )}{b}-\frac {2 a \left (12 a^6-31 a^4 b^2+25 a^2 b^4-6 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d \sqrt {a^2-b^2}}}{b}}{b}}{b \left (a^2-b^2\right )}-\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}\)

Input:

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

Output:

-1/2*(Cos[c + d*x]*Sin[c + d*x]^3)/(b*d*(a + b*Sin[c + d*x])^2) + (-(((4*a 
^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(b*d*(a + b*Sin[c + d*x]))) + (-( 
((((a^2 - b^2)^2*(12*a^2 - b^2)*x)/b - (2*a*(12*a^6 - 31*a^4*b^2 + 25*a^2* 
b^4 - 6*b^6)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b* 
Sqrt[a^2 - b^2]*d))/b + (a*(12*a^2 - 11*b^2)*(a^2 - b^2)*Cos[c + d*x])/(b* 
d))/b) + ((6*a^2 - 5*b^2)*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(b*d))/(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 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 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 3368

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a 
 + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n 
}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || 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 3527

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b 
*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( 
A*d^2*(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, 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 = 1.98 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (6 a^{4}+7 a^{2} b^{2}-10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \left (a^{2}-b^{2}\right )}-\frac {a \,b^{2} \left (19 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} b \left (6 a^{2}-5 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (12 a^{4}-19 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{5}}-\frac {2 \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{2}}{2}+3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3 a b}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (12 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}}{d}\)	\(349\)
default	\(\frac {\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (6 a^{4}+7 a^{2} b^{2}-10 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \left (a^{2}-b^{2}\right )}-\frac {a \,b^{2} \left (19 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} b \left (6 a^{2}-5 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (12 a^{4}-19 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{5}}-\frac {2 \left (\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{2}}{2}+3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3 a b}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (12 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}}{d}\)	\(349\)
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 b^{3} d}-\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {3 a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {i a^{2} \left (-8 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+20 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-17 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-5 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-7 a^{2} b^{2}+6 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) d \,b^{5}}+\frac {6 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}-\frac {19 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}-\frac {6 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}+\frac {19 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}\)	\(773\)

Input:

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

Output:

1/d*(2*a/b^5*((-1/2*a*b^2*(5*a^2-4*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3-1/2 
*b*(6*a^4+7*a^2*b^2-10*b^4)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^2-1/2*a*b^2*(19*a 
^2-16*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a^2*b*(6*a^2-5*b^2)/(a^2-b^2)) 
/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(12*a^4-19*a^2*b^ 
2+6*b^4)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2) 
^(1/2)))-2/b^5*((1/2*tan(1/2*d*x+1/2*c)^3*b^2+3*a*b*tan(1/2*d*x+1/2*c)^2-1 
/2*b^2*tan(1/2*d*x+1/2*c)+3*a*b)/(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))))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1058, normalized size of antiderivative = 3.98 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^2*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

[-1/4*(2*(12*a^6*b^2 - 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 
 8*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^3 - 2*(12*a^8 - 13*a^6*b^2 - 
 11*a^4*b^4 + 13*a^2*b^6 - b^8)*d*x + (12*a^7 - 7*a^5*b^2 - 13*a^3*b^4 + 6 
*a*b^6 - (12*a^5*b^2 - 19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a^6*b 
- 19*a^4*b^3 + 6*a^2*b^5)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^ 
2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin 
(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b* 
sin(d*x + c) - a^2 - b^2)) - 2*(12*a^7*b - 19*a^5*b^3 + 3*a^3*b^5 + 4*a*b^ 
7)*cos(d*x + c) - 2*((a^4*b^4 - 2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 2*(12*a^ 
7*b - 25*a^5*b^3 + 14*a^3*b^5 - a*b^7)*d*x + (18*a^6*b^2 - 36*a^4*b^4 + 19 
*a^2*b^6 - b^8)*cos(d*x + c))*sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)* 
d*cos(d*x + c)^2 - 2*(a^5*b^6 - 2*a^3*b^8 + a*b^10)*d*sin(d*x + c) - (a^6* 
b^5 - a^4*b^7 - a^2*b^9 + b^11)*d), -1/2*((12*a^6*b^2 - 25*a^4*b^4 + 14*a^ 
2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x 
+ c)^3 - (12*a^8 - 13*a^6*b^2 - 11*a^4*b^4 + 13*a^2*b^6 - b^8)*d*x - (12*a 
^7 - 7*a^5*b^2 - 13*a^3*b^4 + 6*a*b^6 - (12*a^5*b^2 - 19*a^3*b^4 + 6*a*b^6 
)*cos(d*x + c)^2 + 2*(12*a^6*b - 19*a^4*b^3 + 6*a^2*b^5)*sin(d*x + c))*sqr 
t(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) 
- (12*a^7*b - 19*a^5*b^3 + 3*a^3*b^5 + 4*a*b^7)*cos(d*x + c) - ((a^4*b^4 - 
 2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 2*(12*a^7*b - 25*a^5*b^3 + 14*a^3*b^...
 

Sympy [F(-1)]

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

integrate(cos(d*x+c)**2*sin(d*x+c)**3/(a+b*sin(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(cos(d*x+c)^2*sin(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

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. 535 vs. \(2 (251) = 502\).

Time = 0.17 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.01 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(2*(12*a^5 - 19*a^3*b^2 + 6*a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*s 
gn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^2*b^5 - 
b^7)*sqrt(a^2 - b^2)) - 2*(6*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 5*a^2*b^3*tan( 
1/2*d*x + 1/2*c)^7 + 12*a^5*tan(1/2*d*x + 1/2*c)^6 + 5*a^3*b^2*tan(1/2*d*x 
 + 1/2*c)^6 - 14*a*b^4*tan(1/2*d*x + 1/2*c)^6 + 54*a^4*b*tan(1/2*d*x + 1/2 
*c)^5 - 45*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 4*b^5*tan(1/2*d*x + 1/2*c)^5 + 
 36*a^5*tan(1/2*d*x + 1/2*c)^4 + 15*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 44*a* 
b^4*tan(1/2*d*x + 1/2*c)^4 + 90*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 87*a^2*b^3* 
tan(1/2*d*x + 1/2*c)^3 + 4*b^5*tan(1/2*d*x + 1/2*c)^3 + 36*a^5*tan(1/2*d*x 
 + 1/2*c)^2 - a^3*b^2*tan(1/2*d*x + 1/2*c)^2 - 30*a*b^4*tan(1/2*d*x + 1/2* 
c)^2 + 42*a^4*b*tan(1/2*d*x + 1/2*c) - 39*a^2*b^3*tan(1/2*d*x + 1/2*c) + 1 
2*a^5 - 11*a^3*b^2)/((a^2*b^4 - b^6)*(a*tan(1/2*d*x + 1/2*c)^4 + 2*b*tan(1 
/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) 
+ a)^2) - (12*a^2 - b^2)*(d*x + c)/b^5)/d
 

Mupad [B] (verification not implemented)

Time = 36.40 (sec) , antiderivative size = 4943, normalized size of antiderivative = 18.58 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

- ((12*a^5 - 11*a^3*b^2)/(b^4*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^7*(6*a^4 
- 5*a^2*b^2))/(b^3*(a^2 - b^2)) - (tan(c/2 + (d*x)/2)^5*(4*b^4 - 54*a^4 + 
45*a^2*b^2))/(b^3*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^3*(90*a^4 + 4*b^4 - 8 
7*a^2*b^2))/(b^3*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^6*(12*a^5 - 14*a*b^4 + 
 5*a^3*b^2))/(b^4*(a^2 - b^2)) - (tan(c/2 + (d*x)/2)^2*(30*a*b^4 - 36*a^5 
+ a^3*b^2))/(b^4*(a^2 - b^2)) + (3*tan(c/2 + (d*x)/2)*(14*a^4 - 13*a^2*b^2 
))/(b^3*(a^2 - b^2)) - (tan(c/2 + (d*x)/2)^4*(11*a*b^2 - 12*a^3)*(3*a^2 + 
4*b^2))/(b^4*(a^2 - b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(4*a^2 + 4*b^2) + tan( 
c/2 + (d*x)/2)^6*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(6*a^2 + 8*b^2) + 
a^2*tan(c/2 + (d*x)/2)^8 + a^2 + 12*a*b*tan(c/2 + (d*x)/2)^3 + 12*a*b*tan( 
c/2 + (d*x)/2)^5 + 4*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2))) 
 - (atan((((a^2*12i - b^2*1i)*((4*(2*a^2*b^12 - 52*a^4*b^10 + 386*a^6*b^8 
- 624*a^8*b^6 + 288*a^10*b^4))/(b^15 - 2*a^2*b^13 + a^4*b^11) - ((a^2*12i 
- b^2*1i)*((4*(4*a*b^16 - 36*a^3*b^14 + 56*a^5*b^12 - 24*a^7*b^10))/(b^15 
- 2*a^2*b^13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(24*a^2*b^16 - 100*a^4*b^ 
14 + 124*a^6*b^12 - 48*a^8*b^10))/(b^16 - 2*a^2*b^14 + a^4*b^12) - ((a^2*1 
2i - b^2*1i)*((4*(8*a^2*b^18 - 16*a^4*b^16 + 8*a^6*b^14))/(b^15 - 2*a^2*b^ 
13 + a^4*b^11) + (8*tan(c/2 + (d*x)/2)*(12*a*b^20 - 32*a^3*b^18 + 28*a^5*b 
^16 - 8*a^7*b^14))/(b^16 - 2*a^2*b^14 + a^4*b^12)))/(2*b^5)))/(2*b^5) + (8 
*tan(c/2 + (d*x)/2)*(2*a*b^14 - 89*a^3*b^12 + 640*a^5*b^10 - 1322*a^7*b...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1054, normalized size of antiderivative = 3.96 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin 
(c + d*x)**2*a**5*b**2 - 76*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b 
)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**3*b**4 + 24*sqrt(a**2 - b**2)*atan 
((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a*b**6 + 96*s 
qrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + 
d*x)*a**6*b - 152*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a** 
2 - b**2))*sin(c + d*x)*a**4*b**3 + 48*sqrt(a**2 - b**2)*atan((tan((c + d* 
x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)*a**2*b**5 + 48*sqrt(a**2 - b* 
*2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**7 - 76*sqrt(a**2 - 
 b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**5*b**2 + 24*sqr 
t(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*a**3*b**4 
+ 2*cos(c + d*x)*sin(c + d*x)**3*a**4*b**4 - 4*cos(c + d*x)*sin(c + d*x)** 
3*a**2*b**6 + 2*cos(c + d*x)*sin(c + d*x)**3*b**8 - 8*cos(c + d*x)*sin(c + 
 d*x)**2*a**5*b**3 + 16*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 - 8*cos(c + 
 d*x)*sin(c + d*x)**2*a*b**7 - 36*cos(c + d*x)*sin(c + d*x)*a**6*b**2 + 70 
*cos(c + d*x)*sin(c + d*x)*a**4*b**4 - 34*cos(c + d*x)*sin(c + d*x)*a**2*b 
**6 - 24*cos(c + d*x)*a**7*b + 46*cos(c + d*x)*a**5*b**3 - 22*cos(c + d*x) 
*a**3*b**5 - 24*sin(c + d*x)**2*a**6*b**2*d*x - 18*sin(c + d*x)**2*a**5*b* 
*3 + 50*sin(c + d*x)**2*a**4*b**4*d*x + 35*sin(c + d*x)**2*a**3*b**5 - 28* 
sin(c + d*x)**2*a**2*b**6*d*x - 17*sin(c + d*x)**2*a*b**7 + 2*sin(c + d...