\(\int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx\) [52]

Optimal result

Integrand size = 22, antiderivative size = 129 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=-\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}+\frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )} \] Output:

-1/2*d*exp(b*x+a)*cos(2*d*x+2*c)/(b^2+4*d^2)+d*exp(b*x+a)*cos(4*d*x+4*c)/( 
2*b^2+32*d^2)+b*exp(b*x+a)*sin(2*d*x+2*c)/(4*b^2+16*d^2)-b*exp(b*x+a)*sin( 
4*d*x+4*c)/(8*b^2+128*d^2)
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\frac {1}{8} e^{a+b x} \left (\frac {2 (-2 d \cos (2 (c+d x))+b \sin (2 (c+d x)))}{b^2+4 d^2}+\frac {4 d \cos (4 (c+d x))-b \sin (4 (c+d x))}{b^2+16 d^2}\right ) \] Input:

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

Output:

(E^(a + b*x)*((2*(-2*d*Cos[2*(c + d*x)] + b*Sin[2*(c + d*x)]))/(b^2 + 4*d^ 
2) + (4*d*Cos[4*(c + d*x)] - b*Sin[4*(c + d*x)])/(b^2 + 16*d^2)))/8
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4972, 2009}

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.

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

\(\Big \downarrow \) 4972

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b e^{a+b x} \sin (2 c+2 d x)}{4 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (4 c+4 d x)}{8 \left (b^2+16 d^2\right )}-\frac {d e^{a+b x} \cos (2 c+2 d x)}{2 \left (b^2+4 d^2\right )}+\frac {d e^{a+b x} \cos (4 c+4 d x)}{2 \left (b^2+16 d^2\right )}\)

Input:

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

Output:

-1/2*(d*E^(a + b*x)*Cos[2*c + 2*d*x])/(b^2 + 4*d^2) + (d*E^(a + b*x)*Cos[4 
*c + 4*d*x])/(2*(b^2 + 16*d^2)) + (b*E^(a + b*x)*Sin[2*c + 2*d*x])/(4*(b^2 
 + 4*d^2)) - (b*E^(a + b*x)*Sin[4*c + 4*d*x])/(8*(b^2 + 16*d^2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4972

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_ 
.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)), 
Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] 
 && IGtQ[m, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(-\frac {\left (\left (b^{3}+4 b \,d^{2}\right ) \sin \left (4 d x +4 c \right )+\left (-4 b^{2} d -16 d^{3}\right ) \cos \left (4 d x +4 c \right )-2 \left (b^{2}+16 d^{2}\right ) \left (-2 d \cos \left (2 d x +2 c \right )+b \sin \left (2 d x +2 c \right )\right )\right ) {\mathrm e}^{b x +a}}{8 b^{4}+160 b^{2} d^{2}+512 d^{4}}\)	\(108\)
default	\(-\frac {d \,{\mathrm e}^{b x +a} \cos \left (2 d x +2 c \right )}{2 \left (b^{2}+4 d^{2}\right )}+\frac {b \,{\mathrm e}^{b x +a} \sin \left (2 d x +2 c \right )}{4 b^{2}+16 d^{2}}+\frac {d \,{\mathrm e}^{b x +a} \cos \left (4 d x +4 c \right )}{2 b^{2}+32 d^{2}}-\frac {b \,{\mathrm e}^{b x +a} \sin \left (4 d x +4 c \right )}{8 \left (b^{2}+16 d^{2}\right )}\)	\(118\)
risch	\(\frac {i {\mathrm e}^{b x +a} \left (-8 i d \left (b^{2}+4 d^{2}\right ) \cos \left (4 d x +4 c \right )-i \left (-2 b^{3}-8 b \,d^{2}\right ) \sin \left (4 d x +4 c \right )+8 i d \left (b^{2}+16 d^{2}\right ) \cos \left (2 d x +2 c \right )-i \left (4 b^{3}+64 b \,d^{2}\right ) \sin \left (2 d x +2 c \right )\right )}{16 \left (4 i d +b \right ) \left (2 i d +b \right ) \left (2 i d -b \right ) \left (4 i d -b \right )}\)	\(139\)
norman	\(\frac {-\frac {6 d^{3} {\mathrm e}^{b x +a}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {6 d^{3} {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {12 b \,d^{2} {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {12 b \,d^{2} {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {4 b \left (2 b^{2}+11 d^{2}\right ) {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {4 b \left (2 b^{2}+11 d^{2}\right ) {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {12 d \left (b^{2}+2 d^{2}\right ) {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {12 d \left (b^{2}+2 d^{2}\right ) {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {20 d \left (2 b^{2}+11 d^{2}\right ) {\mathrm e}^{b x +a} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\)	\(417\)
orering	\(\frac {4 b \left (b^{2}+10 d^{2}\right ) {\mathrm e}^{b x +a} \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {2 \left (3 b^{2}+10 d^{2}\right ) \left (b \,{\mathrm e}^{b x +a} \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-{\mathrm e}^{b x +a} d \sin \left (d x +c \right )^{4}+3 \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} d \right )}{b^{4}+20 b^{2} d^{2}+64 d^{4}}+\frac {4 b \left (b^{2} {\mathrm e}^{b x +a} \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-2 b \,{\mathrm e}^{b x +a} d \sin \left (d x +c \right )^{4}+6 b \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} d -10 \,{\mathrm e}^{b x +a} d^{2} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )+6 \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) d^{2}\right )}{b^{4}+20 b^{2} d^{2}+64 d^{4}}-\frac {b^{3} {\mathrm e}^{b x +a} \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-3 b^{2} {\mathrm e}^{b x +a} d \sin \left (d x +c \right )^{4}+9 b^{2} {\mathrm e}^{b x +a} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} d -30 b \,{\mathrm e}^{b x +a} d^{2} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )+18 b \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) d^{2}-48 \,{\mathrm e}^{b x +a} d^{3} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}+10 \,{\mathrm e}^{b x +a} d^{3} \sin \left (d x +c \right )^{4}+6 \,{\mathrm e}^{b x +a} \cos \left (d x +c \right )^{4} d^{3}}{b^{4}+20 b^{2} d^{2}+64 d^{4}}\)	\(502\)

Input:

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

Output:

-((b^3+4*b*d^2)*sin(4*d*x+4*c)+(-4*b^2*d-16*d^3)*cos(4*d*x+4*c)-2*(b^2+16* 
d^2)*(-2*d*cos(2*d*x+2*c)+b*sin(2*d*x+2*c)))*exp(b*x+a)/(8*b^4+160*b^2*d^2 
+512*d^4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=-\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (b^{3} + 10 \, b d^{2}\right )} \cos \left (d x + c\right )\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - {\left (4 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + b^{2} d + 10 \, d^{3} - {\left (5 \, b^{2} d + 32 \, d^{3}\right )} \cos \left (d x + c\right )^{2}\right )} e^{\left (b x + a\right )}}{b^{4} + 20 \, b^{2} d^{2} + 64 \, d^{4}} \] Input:

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

Output:

-(((b^3 + 4*b*d^2)*cos(d*x + c)^3 - (b^3 + 10*b*d^2)*cos(d*x + c))*e^(b*x 
+ a)*sin(d*x + c) - (4*(b^2*d + 4*d^3)*cos(d*x + c)^4 + b^2*d + 10*d^3 - ( 
5*b^2*d + 32*d^3)*cos(d*x + c)^2)*e^(b*x + a))/(b^4 + 20*b^2*d^2 + 64*d^4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.57 (sec) , antiderivative size = 1357, normalized size of antiderivative = 10.52 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((x*exp(a)*sin(c)**3*cos(c), Eq(b, 0) & Eq(d, 0)), (I*x*exp(a)*ex 
p(-4*I*d*x)*sin(c + d*x)**4/16 + x*exp(a)*exp(-4*I*d*x)*sin(c + d*x)**3*co 
s(c + d*x)/4 - 3*I*x*exp(a)*exp(-4*I*d*x)*sin(c + d*x)**2*cos(c + d*x)**2/ 
8 - x*exp(a)*exp(-4*I*d*x)*sin(c + d*x)*cos(c + d*x)**3/4 + I*x*exp(a)*exp 
(-4*I*d*x)*cos(c + d*x)**4/16 - exp(a)*exp(-4*I*d*x)*sin(c + d*x)**4/(24*d 
) + 11*I*exp(a)*exp(-4*I*d*x)*sin(c + d*x)**3*cos(c + d*x)/(48*d) + 5*I*ex 
p(a)*exp(-4*I*d*x)*sin(c + d*x)*cos(c + d*x)**3/(48*d) + exp(a)*exp(-4*I*d 
*x)*cos(c + d*x)**4/(24*d), Eq(b, -4*I*d)), (I*x*exp(a)*exp(-2*I*d*x)*sin( 
c + d*x)**4/8 + x*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**3*cos(c + d*x)/4 + x* 
exp(a)*exp(-2*I*d*x)*sin(c + d*x)*cos(c + d*x)**3/4 - I*x*exp(a)*exp(-2*I* 
d*x)*cos(c + d*x)**4/8 + 7*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**4/(48*d) - I 
*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**3*cos(c + d*x)/(6*d) - exp(a)*exp(-2*I 
*d*x)*sin(c + d*x)**2*cos(c + d*x)**2/(4*d) - exp(a)*exp(-2*I*d*x)*cos(c + 
 d*x)**4/(16*d), Eq(b, -2*I*d)), (-I*x*exp(a)*exp(2*I*d*x)*sin(c + d*x)**4 
/8 + x*exp(a)*exp(2*I*d*x)*sin(c + d*x)**3*cos(c + d*x)/4 + x*exp(a)*exp(2 
*I*d*x)*sin(c + d*x)*cos(c + d*x)**3/4 + I*x*exp(a)*exp(2*I*d*x)*cos(c + d 
*x)**4/8 + 7*exp(a)*exp(2*I*d*x)*sin(c + d*x)**4/(48*d) + I*exp(a)*exp(2*I 
*d*x)*sin(c + d*x)**3*cos(c + d*x)/(6*d) - exp(a)*exp(2*I*d*x)*sin(c + d*x 
)**2*cos(c + d*x)**2/(4*d) - exp(a)*exp(2*I*d*x)*cos(c + d*x)**4/(16*d), E 
q(b, 2*I*d)), (-I*x*exp(a)*exp(4*I*d*x)*sin(c + d*x)**4/16 + x*exp(a)*e...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (117) = 234\).

Time = 0.09 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.26 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx =\text {Too large to display} \] Input:

integrate(exp(b*x+a)*cos(d*x+c)*sin(d*x+c)^3,x, algorithm="maxima")
 

Output:

1/16*((4*b^2*d*cos(4*c)*e^a + 16*d^3*cos(4*c)*e^a - b^3*e^a*sin(4*c) - 4*b 
*d^2*e^a*sin(4*c))*cos(4*d*x)*e^(b*x) + (4*b^2*d*cos(4*c)*e^a + 16*d^3*cos 
(4*c)*e^a + b^3*e^a*sin(4*c) + 4*b*d^2*e^a*sin(4*c))*cos(4*d*x + 8*c)*e^(b 
*x) - 2*(2*b^2*d*cos(4*c)*e^a + 32*d^3*cos(4*c)*e^a + b^3*e^a*sin(4*c) + 1 
6*b*d^2*e^a*sin(4*c))*cos(2*d*x + 6*c)*e^(b*x) - 2*(2*b^2*d*cos(4*c)*e^a + 
 32*d^3*cos(4*c)*e^a - b^3*e^a*sin(4*c) - 16*b*d^2*e^a*sin(4*c))*cos(2*d*x 
 - 2*c)*e^(b*x) - (b^3*cos(4*c)*e^a + 4*b*d^2*cos(4*c)*e^a + 4*b^2*d*e^a*s 
in(4*c) + 16*d^3*e^a*sin(4*c))*e^(b*x)*sin(4*d*x) - (b^3*cos(4*c)*e^a + 4* 
b*d^2*cos(4*c)*e^a - 4*b^2*d*e^a*sin(4*c) - 16*d^3*e^a*sin(4*c))*e^(b*x)*s 
in(4*d*x + 8*c) + 2*(b^3*cos(4*c)*e^a + 16*b*d^2*cos(4*c)*e^a - 2*b^2*d*e^ 
a*sin(4*c) - 32*d^3*e^a*sin(4*c))*e^(b*x)*sin(2*d*x + 6*c) + 2*(b^3*cos(4* 
c)*e^a + 16*b*d^2*cos(4*c)*e^a + 2*b^2*d*e^a*sin(4*c) + 32*d^3*e^a*sin(4*c 
))*e^(b*x)*sin(2*d*x - 2*c))/(b^4*cos(4*c)^2 + b^4*sin(4*c)^2 + 64*(cos(4* 
c)^2 + sin(4*c)^2)*d^4 + 20*(b^2*cos(4*c)^2 + b^2*sin(4*c)^2)*d^2)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\frac {1}{8} \, {\left (\frac {4 \, d \cos \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}} - \frac {b \sin \left (4 \, d x + 4 \, c\right )}{b^{2} + 16 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {1}{4} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \] Input:

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

Output:

1/8*(4*d*cos(4*d*x + 4*c)/(b^2 + 16*d^2) - b*sin(4*d*x + 4*c)/(b^2 + 16*d^ 
2))*e^(b*x + a) - 1/4*(2*d*cos(2*d*x + 2*c)/(b^2 + 4*d^2) - b*sin(2*d*x + 
2*c)/(b^2 + 4*d^2))*e^(b*x + a)
 

Mupad [B] (verification not implemented)

Time = 21.00 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{8\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (4\,d\,x\right )-\sin \left (4\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,c\right )-\sin \left (4\,c\right )\,1{}\mathrm {i}\right )}{16\,\left (4\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (4\,d\,x\right )+\sin \left (4\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (4\,c\right )+\sin \left (4\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,\left (b+d\,4{}\mathrm {i}\right )} \] Input:

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

Output:

(exp(a + b*x)*(cos(4*d*x) - sin(4*d*x)*1i)*(cos(4*c) - sin(4*c)*1i))/(16*( 
b*1i + 4*d)) - (exp(a + b*x)*(cos(2*d*x) - sin(2*d*x)*1i)*(cos(2*c) - sin( 
2*c)*1i))/(8*(b*1i + 2*d)) - (exp(a + b*x)*(cos(2*d*x) + sin(2*d*x)*1i)*(c 
os(2*c) + sin(2*c)*1i)*1i)/(8*(b + d*2i)) + (exp(a + b*x)*(cos(4*d*x) + si 
n(4*d*x)*1i)*(cos(4*c) + sin(4*c)*1i)*1i)/(16*(b + d*4i))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.01 \[ \int e^{a+b x} \cos (c+d x) \sin ^3(c+d x) \, dx=\frac {e^{b x +a} \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3}+4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b \,d^{2}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b \,d^{2}+4 \sin \left (d x +c \right )^{4} b^{2} d +16 \sin \left (d x +c \right )^{4} d^{3}-3 \sin \left (d x +c \right )^{2} b^{2} d -6 d^{3}\right )}{b^{4}+20 b^{2} d^{2}+64 d^{4}} \] Input:

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

Output:

(e**(a + b*x)*(cos(c + d*x)*sin(c + d*x)**3*b**3 + 4*cos(c + d*x)*sin(c + 
d*x)**3*b*d**2 + 6*cos(c + d*x)*sin(c + d*x)*b*d**2 + 4*sin(c + d*x)**4*b* 
*2*d + 16*sin(c + d*x)**4*d**3 - 3*sin(c + d*x)**2*b**2*d - 6*d**3))/(b**4 
 + 20*b**2*d**2 + 64*d**4)