\(\int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [263]

Optimal result

Integrand size = 28, antiderivative size = 219 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 f^3 x}{8 a d^3}+\frac {(e+f x)^3}{4 a d}-\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {(e+f x)^3 \sin (c+d x)}{a d}+\frac {3 f^3 \cos (c+d x) \sin (c+d x)}{8 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a d^2}+\frac {3 f^2 (e+f x) \sin ^2(c+d x)}{4 a d^3}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 a d} \] Output:

-3/8*f^3*x/a/d^3+1/4*(f*x+e)^3/a/d-6*f^3*cos(d*x+c)/a/d^4+3*f*(f*x+e)^2*co 
s(d*x+c)/a/d^2-6*f^2*(f*x+e)*sin(d*x+c)/a/d^3+(f*x+e)^3*sin(d*x+c)/a/d+3/8 
*f^3*cos(d*x+c)*sin(d*x+c)/a/d^4-3/4*f*(f*x+e)^2*cos(d*x+c)*sin(d*x+c)/a/d 
^2+3/4*f^2*(f*x+e)*sin(d*x+c)^2/a/d^3-1/2*(f*x+e)^3*sin(d*x+c)^2/a/d
 

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.60 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {96 f \left (-2 f^2+d^2 (e+f x)^2\right ) \cos (c+d x)+4 d (e+f x) \left (-3 f^2+2 d^2 (e+f x)^2\right ) \cos (2 (c+d x))+4 \left (8 d (e+f x) \left (-6 f^2+d^2 (e+f x)^2\right )-3 f \left (-f^2+2 d^2 (e+f x)^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{32 a d^4} \] Input:

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

Output:

(96*f*(-2*f^2 + d^2*(e + f*x)^2)*Cos[c + d*x] + 4*d*(e + f*x)*(-3*f^2 + 2* 
d^2*(e + f*x)^2)*Cos[2*(c + d*x)] + 4*(8*d*(e + f*x)*(-6*f^2 + d^2*(e + f* 
x)^2) - 3*f*(-f^2 + 2*d^2*(e + f*x)^2)*Cos[c + d*x])*Sin[c + d*x])/(32*a*d 
^4)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5034, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118, 4904, 3042, 3792, 17, 3042, 3115, 24}

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.

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

\(\Big \downarrow \) 5034

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 4904

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \int \sin ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \int \sin (c+d x)^2dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (-\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}}{a}-\frac {\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {3 f \left (\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{a}\)

Input:

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

Output:

(((e + f*x)^3*Sin[c + d*x])/d - (3*f*(-(((e + f*x)^2*Cos[c + d*x])/d) + (2 
*f*((f*Cos[c + d*x])/d^2 + ((e + f*x)*Sin[c + d*x])/d))/d))/d)/a - (((e + 
f*x)^3*Sin[c + d*x]^2)/(2*d) - (3*f*((e + f*x)^3/(6*f) - ((e + f*x)^2*Cos[ 
c + d*x]*Sin[c + d*x])/(2*d) + (f*(e + f*x)*Sin[c + d*x]^2)/(2*d^2) - (f^2 
*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/(2*d^2)))/(2*d))/a
 

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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 3042

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

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 4904

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x 
_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) 
, x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 5034

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

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.73

method	result	size
parallelrisch	\(\frac {2 \left (f x +e \right ) \left (\left (f x +e \right )^{2} d^{2}-\frac {3 f^{2}}{2}\right ) d \cos \left (2 d x +2 c \right )-3 \left (\left (f x +e \right )^{2} d^{2}-\frac {f^{2}}{2}\right ) f \sin \left (2 d x +2 c \right )+8 \left (\left (f x +e \right )^{2} d^{2}-6 f^{2}\right ) \left (f x +e \right ) d \sin \left (d x +c \right )+24 \left (\left (f x +e \right )^{2} d^{2}-2 f^{2}\right ) f \cos \left (d x +c \right )-2 d^{3} e^{3}+24 d^{2} e^{2} f +3 d e \,f^{2}-48 f^{3}}{8 d^{4} a}\)	\(159\)
risch	\(\frac {3 f \left (d^{2} x^{2} f^{2}+2 e f x \,d^{2}+d^{2} e^{2}-2 f^{2}\right ) \cos \left (d x +c \right )}{d^{4} a}+\frac {\left (d^{2} x^{3} f^{3}+3 d^{2} e \,f^{2} x^{2}+3 d^{2} e^{2} f x +d^{2} e^{3}-6 f^{3} x -6 e \,f^{2}\right ) \sin \left (d x +c \right )}{d^{3} a}+\frac {\left (2 d^{2} x^{3} f^{3}+6 d^{2} e \,f^{2} x^{2}+6 d^{2} e^{2} f x +2 d^{2} e^{3}-3 f^{3} x -3 e \,f^{2}\right ) \cos \left (2 d x +2 c \right )}{8 d^{3} a}-\frac {3 f \left (2 d^{2} x^{2} f^{2}+4 e f x \,d^{2}+2 d^{2} e^{2}-f^{2}\right ) \sin \left (2 d x +2 c \right )}{16 d^{4} a}\)	\(235\)
derivativedivides	\(\frac {-\frac {c^{3} f^{3} \cos \left (d x +c \right )^{2}}{2}+\frac {3 c^{2} d e \,f^{2} \cos \left (d x +c \right )^{2}}{2}-3 c^{2} f^{3} \left (-\frac {\left (d x +c \right ) \cos \left (d x +c \right )^{2}}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\frac {3 c \,d^{2} e^{2} f \cos \left (d x +c \right )^{2}}{2}+6 c d e \,f^{2} \left (-\frac {\left (d x +c \right ) \cos \left (d x +c \right )^{2}}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+3 c \,f^{3} \left (-\frac {\left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\sin \left (d x +c \right )^{2}}{4}\right )+\frac {d^{3} e^{3} \cos \left (d x +c \right )^{2}}{2}-3 d^{2} e^{2} f \left (-\frac {\left (d x +c \right ) \cos \left (d x +c \right )^{2}}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-3 d e \,f^{2} \left (-\frac {\left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\sin \left (d x +c \right )^{2}}{4}\right )-f^{3} \left (-\frac {\left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}{2}+\frac {3 \left (d x +c \right )^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 \left (d x +c \right ) \cos \left (d x +c \right )^{2}}{4}-\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}-\frac {\left (d x +c \right )^{3}}{2}\right )-\sin \left (d x +c \right ) c^{3} f^{3}+3 \sin \left (d x +c \right ) c^{2} d e \,f^{2}+3 c^{2} f^{3} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 \sin \left (d x +c \right ) c \,d^{2} e^{2} f -6 c d e \,f^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 c \,f^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+\sin \left (d x +c \right ) d^{3} e^{3}+3 d^{2} e^{2} f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+f^{3} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4} a}\)	\(736\)
default	\(\frac {-\frac {c^{3} f^{3} \cos \left (d x +c \right )^{2}}{2}+\frac {3 c^{2} d e \,f^{2} \cos \left (d x +c \right )^{2}}{2}-3 c^{2} f^{3} \left (-\frac {\left (d x +c \right ) \cos \left (d x +c \right )^{2}}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-\frac {3 c \,d^{2} e^{2} f \cos \left (d x +c \right )^{2}}{2}+6 c d e \,f^{2} \left (-\frac {\left (d x +c \right ) \cos \left (d x +c \right )^{2}}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )+3 c \,f^{3} \left (-\frac {\left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\sin \left (d x +c \right )^{2}}{4}\right )+\frac {d^{3} e^{3} \cos \left (d x +c \right )^{2}}{2}-3 d^{2} e^{2} f \left (-\frac {\left (d x +c \right ) \cos \left (d x +c \right )^{2}}{2}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{4}+\frac {d x}{4}+\frac {c}{4}\right )-3 d e \,f^{2} \left (-\frac {\left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}{2}+\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {\left (d x +c \right )^{2}}{4}-\frac {\sin \left (d x +c \right )^{2}}{4}\right )-f^{3} \left (-\frac {\left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}{2}+\frac {3 \left (d x +c \right )^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {3 \left (d x +c \right ) \cos \left (d x +c \right )^{2}}{4}-\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}-\frac {\left (d x +c \right )^{3}}{2}\right )-\sin \left (d x +c \right ) c^{3} f^{3}+3 \sin \left (d x +c \right ) c^{2} d e \,f^{2}+3 c^{2} f^{3} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 \sin \left (d x +c \right ) c \,d^{2} e^{2} f -6 c d e \,f^{2} \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )-3 c \,f^{3} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+\sin \left (d x +c \right ) d^{3} e^{3}+3 d^{2} e^{2} f \left (\cos \left (d x +c \right )+\left (d x +c \right ) \sin \left (d x +c \right )\right )+3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sin \left (d x +c \right )-2 \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \left (d x +c \right )\right )+f^{3} \left (\left (d x +c \right )^{3} \sin \left (d x +c \right )+3 \left (d x +c \right )^{2} \cos \left (d x +c \right )-6 \cos \left (d x +c \right )-6 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4} a}\)	\(736\)

Input:

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

Output:

1/8*(2*(f*x+e)*((f*x+e)^2*d^2-3/2*f^2)*d*cos(2*d*x+2*c)-3*((f*x+e)^2*d^2-1 
/2*f^2)*f*sin(2*d*x+2*c)+8*((f*x+e)^2*d^2-6*f^2)*(f*x+e)*d*sin(d*x+c)+24*( 
(f*x+e)^2*d^2-2*f^2)*f*cos(d*x+c)-2*d^3*e^3+24*d^2*e^2*f+3*d*e*f^2-48*f^3) 
/d^4/a
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} - 2 \, {\left (2 \, d^{3} f^{3} x^{3} + 6 \, d^{3} e f^{2} x^{2} + 2 \, d^{3} e^{3} - 3 \, d e f^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, d^{3} e^{2} f - d f^{3}\right )} x - 24 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + d^{2} e^{2} f - 2 \, f^{3}\right )} \cos \left (d x + c\right ) - {\left (8 \, d^{3} f^{3} x^{3} + 24 \, d^{3} e f^{2} x^{2} + 8 \, d^{3} e^{3} - 48 \, d e f^{2} + 24 \, {\left (d^{3} e^{2} f - 2 \, d f^{3}\right )} x - 3 \, {\left (2 \, d^{2} f^{3} x^{2} + 4 \, d^{2} e f^{2} x + 2 \, d^{2} e^{2} f - f^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, a d^{4}} \] Input:

integrate((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
 

Output:

-1/8*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 - 2*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 
 + 2*d^3*e^3 - 3*d*e*f^2 + 3*(2*d^3*e^2*f - d*f^3)*x)*cos(d*x + c)^2 + 3*( 
2*d^3*e^2*f - d*f^3)*x - 24*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f - 2*f 
^3)*cos(d*x + c) - (8*d^3*f^3*x^3 + 24*d^3*e*f^2*x^2 + 8*d^3*e^3 - 48*d*e* 
f^2 + 24*(d^3*e^2*f - 2*d*f^3)*x - 3*(2*d^2*f^3*x^2 + 4*d^2*e*f^2*x + 2*d^ 
2*e^2*f - f^3)*cos(d*x + c))*sin(d*x + c))/(a*d^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2725 vs. \(2 (204) = 408\).

Time = 4.61 (sec) , antiderivative size = 2725, normalized size of antiderivative = 12.44 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)**3*cos(d*x+c)**3/(a+a*sin(d*x+c)),x)
                                                                                    
                                                                                    
 

Output:

Piecewise((16*d**3*e**3*tan(c/2 + d*x/2)**3/(8*a*d**4*tan(c/2 + d*x/2)**4 
+ 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 16*d**3*e**3*tan(c/2 + d*x/2 
)**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d 
**4) + 16*d**3*e**3*tan(c/2 + d*x/2)/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a* 
d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e**2*f*x*tan(c/2 + d*x/2)**4 
/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) 
 + 48*d**3*e**2*f*x*tan(c/2 + d*x/2)**3/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16 
*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 36*d**3*e**2*f*x*tan(c/2 + d*x/2 
)**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d 
**4) + 48*d**3*e**2*f*x*tan(c/2 + d*x/2)/(8*a*d**4*tan(c/2 + d*x/2)**4 + 1 
6*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e**2*f*x/(8*a*d**4*tan(c 
/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 6*d**3*e*f**2 
*x**2*tan(c/2 + d*x/2)**4/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/ 
2 + d*x/2)**2 + 8*a*d**4) + 48*d**3*e*f**2*x**2*tan(c/2 + d*x/2)**3/(8*a*d 
**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) - 36*d 
**3*e*f**2*x**2*tan(c/2 + d*x/2)**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d 
**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) + 48*d**3*e*f**2*x**2*tan(c/2 + d*x/2) 
/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + d*x/2)**2 + 8*a*d**4) 
 + 6*d**3*e*f**2*x**2/(8*a*d**4*tan(c/2 + d*x/2)**4 + 16*a*d**4*tan(c/2 + 
d*x/2)**2 + 8*a*d**4) + 2*d**3*f**3*x**3*tan(c/2 + d*x/2)**4/(8*a*d**4*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (207) = 414\).

Time = 0.07 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.61 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
 

Output:

-1/16*(8*(sin(d*x + c)^2 - 2*sin(d*x + c))*e^3/a - 24*(sin(d*x + c)^2 - 2* 
sin(d*x + c))*c*e^2*f/(a*d) + 24*(sin(d*x + c)^2 - 2*sin(d*x + c))*c^2*e*f 
^2/(a*d^2) - 8*(sin(d*x + c)^2 - 2*sin(d*x + c))*c^3*f^3/(a*d^3) - 6*(2*(d 
*x + c)*cos(2*d*x + 2*c) + 8*(d*x + c)*sin(d*x + c) + 8*cos(d*x + c) - sin 
(2*d*x + 2*c))*e^2*f/(a*d) + 12*(2*(d*x + c)*cos(2*d*x + 2*c) + 8*(d*x + c 
)*sin(d*x + c) + 8*cos(d*x + c) - sin(2*d*x + 2*c))*c*e*f^2/(a*d^2) - 6*(2 
*(d*x + c)*cos(2*d*x + 2*c) + 8*(d*x + c)*sin(d*x + c) + 8*cos(d*x + c) - 
sin(2*d*x + 2*c))*c^2*f^3/(a*d^3) - 6*((2*(d*x + c)^2 - 1)*cos(2*d*x + 2*c 
) + 16*(d*x + c)*cos(d*x + c) - 2*(d*x + c)*sin(2*d*x + 2*c) + 8*((d*x + c 
)^2 - 2)*sin(d*x + c))*e*f^2/(a*d^2) + 6*((2*(d*x + c)^2 - 1)*cos(2*d*x + 
2*c) + 16*(d*x + c)*cos(d*x + c) - 2*(d*x + c)*sin(2*d*x + 2*c) + 8*((d*x 
+ c)^2 - 2)*sin(d*x + c))*c*f^3/(a*d^3) - (2*(2*(d*x + c)^3 - 3*d*x - 3*c) 
*cos(2*d*x + 2*c) + 48*((d*x + c)^2 - 2)*cos(d*x + c) - 3*(2*(d*x + c)^2 - 
 1)*sin(2*d*x + 2*c) + 16*((d*x + c)^3 - 6*d*x - 6*c)*sin(d*x + c))*f^3/(a 
*d^3))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3893 vs. \(2 (207) = 414\).

Time = 0.30 (sec) , antiderivative size = 3893, normalized size of antiderivative = 17.78 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(2*d^3*f^3*x^3*tan(1/2*d*x)^4*tan(1/2*c)^4 - 16*d^3*f^3*x^3*tan(1/2*d* 
x)^4*tan(1/2*c)^3 - 16*d^3*f^3*x^3*tan(1/2*d*x)^3*tan(1/2*c)^4 + 6*d^3*e*f 
^2*x^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 12*d^3*f^3*x^3*tan(1/2*d*x)^4*tan(1/2 
*c)^2 - 32*d^3*f^3*x^3*tan(1/2*d*x)^3*tan(1/2*c)^3 - 48*d^3*e*f^2*x^2*tan( 
1/2*d*x)^4*tan(1/2*c)^3 - 12*d^3*f^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^4 - 48* 
d^3*e*f^2*x^2*tan(1/2*d*x)^3*tan(1/2*c)^4 + 6*d^3*e^2*f*x*tan(1/2*d*x)^4*t 
an(1/2*c)^4 + 24*d^2*f^3*x^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 16*d^3*f^3*x^3* 
tan(1/2*d*x)^4*tan(1/2*c) - 36*d^3*e*f^2*x^2*tan(1/2*d*x)^4*tan(1/2*c)^2 - 
 96*d^3*e*f^2*x^2*tan(1/2*d*x)^3*tan(1/2*c)^3 - 48*d^3*e^2*f*x*tan(1/2*d*x 
)^4*tan(1/2*c)^3 + 12*d^2*f^3*x^2*tan(1/2*d*x)^4*tan(1/2*c)^3 - 16*d^3*f^3 
*x^3*tan(1/2*d*x)*tan(1/2*c)^4 - 36*d^3*e*f^2*x^2*tan(1/2*d*x)^2*tan(1/2*c 
)^4 - 48*d^3*e^2*f*x*tan(1/2*d*x)^3*tan(1/2*c)^4 + 12*d^2*f^3*x^2*tan(1/2* 
d*x)^3*tan(1/2*c)^4 + 2*d^3*e^3*tan(1/2*d*x)^4*tan(1/2*c)^4 + 48*d^2*e*f^2 
*x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 2*d^3*f^3*x^3*tan(1/2*d*x)^4 + 32*d^3*f^3 
*x^3*tan(1/2*d*x)^3*tan(1/2*c) - 48*d^3*e*f^2*x^2*tan(1/2*d*x)^4*tan(1/2*c 
) + 72*d^3*f^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - 36*d^3*e^2*f*x*tan(1/2*d* 
x)^4*tan(1/2*c)^2 + 32*d^3*f^3*x^3*tan(1/2*d*x)*tan(1/2*c)^3 - 96*d^3*e^2* 
f*x*tan(1/2*d*x)^3*tan(1/2*c)^3 - 96*d^2*f^3*x^2*tan(1/2*d*x)^3*tan(1/2*c) 
^3 - 16*d^3*e^3*tan(1/2*d*x)^4*tan(1/2*c)^3 + 24*d^2*e*f^2*x*tan(1/2*d*x)^ 
4*tan(1/2*c)^3 + 2*d^3*f^3*x^3*tan(1/2*c)^4 - 48*d^3*e*f^2*x^2*tan(1/2*...
 

Mupad [B] (verification not implemented)

Time = 38.76 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {3\,f^3\,\sin \left (2\,c+2\,d\,x\right )}{2}-48\,f^3\,\cos \left (c+d\,x\right )+8\,d^3\,e^3\,\sin \left (c+d\,x\right )+2\,d^3\,e^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,e^2\,f\,\sin \left (2\,c+2\,d\,x\right )+24\,d^2\,f^3\,x^2\,\cos \left (c+d\,x\right )+8\,d^3\,f^3\,x^3\,\sin \left (c+d\,x\right )-48\,d\,e\,f^2\,\sin \left (c+d\,x\right )-48\,d\,f^3\,x\,\sin \left (c+d\,x\right )+2\,d^3\,f^3\,x^3\,\cos \left (2\,c+2\,d\,x\right )-3\,d^2\,f^3\,x^2\,\sin \left (2\,c+2\,d\,x\right )-3\,d\,e\,f^2\,\cos \left (2\,c+2\,d\,x\right )+24\,d^2\,e^2\,f\,\cos \left (c+d\,x\right )-3\,d\,f^3\,x\,\cos \left (2\,c+2\,d\,x\right )+48\,d^2\,e\,f^2\,x\,\cos \left (c+d\,x\right )+24\,d^3\,e^2\,f\,x\,\sin \left (c+d\,x\right )+6\,d^3\,e^2\,f\,x\,\cos \left (2\,c+2\,d\,x\right )-6\,d^2\,e\,f^2\,x\,\sin \left (2\,c+2\,d\,x\right )+24\,d^3\,e\,f^2\,x^2\,\sin \left (c+d\,x\right )+6\,d^3\,e\,f^2\,x^2\,\cos \left (2\,c+2\,d\,x\right )}{8\,a\,d^4} \] Input:

int((cos(c + d*x)^3*(e + f*x)^3)/(a + a*sin(c + d*x)),x)
 

Output:

((3*f^3*sin(2*c + 2*d*x))/2 - 48*f^3*cos(c + d*x) + 8*d^3*e^3*sin(c + d*x) 
 + 2*d^3*e^3*cos(2*c + 2*d*x) - 3*d^2*e^2*f*sin(2*c + 2*d*x) + 24*d^2*f^3* 
x^2*cos(c + d*x) + 8*d^3*f^3*x^3*sin(c + d*x) - 48*d*e*f^2*sin(c + d*x) - 
48*d*f^3*x*sin(c + d*x) + 2*d^3*f^3*x^3*cos(2*c + 2*d*x) - 3*d^2*f^3*x^2*s 
in(2*c + 2*d*x) - 3*d*e*f^2*cos(2*c + 2*d*x) + 24*d^2*e^2*f*cos(c + d*x) - 
 3*d*f^3*x*cos(2*c + 2*d*x) + 48*d^2*e*f^2*x*cos(c + d*x) + 24*d^3*e^2*f*x 
*sin(c + d*x) + 6*d^3*e^2*f*x*cos(2*c + 2*d*x) - 6*d^2*e*f^2*x*sin(2*c + 2 
*d*x) + 24*d^3*e*f^2*x^2*sin(c + d*x) + 6*d^3*e*f^2*x^2*cos(2*c + 2*d*x))/ 
(8*a*d^4)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.83 \[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-12 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d^{2} e \,f^{2} x +8 d^{3} e^{3}+2 d^{3} f^{3} x^{3}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) f^{3}-4 \sin \left (d x +c \right )^{2} d^{3} e^{3}+8 \sin \left (d x +c \right ) d^{3} e^{3}+6 d^{3} e^{2} f x +6 d^{3} e \,f^{2} x^{2}-3 d \,f^{3} x -48 \cos \left (d x +c \right ) f^{3}-12 d e \,f^{2}+24 \cos \left (d x +c \right ) d^{2} e^{2} f +24 \cos \left (d x +c \right ) d^{2} f^{3} x^{2}-4 \sin \left (d x +c \right )^{2} d^{3} f^{3} x^{3}+6 \sin \left (d x +c \right )^{2} d e \,f^{2}+6 \sin \left (d x +c \right )^{2} d \,f^{3} x +8 \sin \left (d x +c \right ) d^{3} f^{3} x^{3}-48 \sin \left (d x +c \right ) d e \,f^{2}-48 \sin \left (d x +c \right ) d \,f^{3} x -6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d^{2} e^{2} f -6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d^{2} f^{3} x^{2}+48 \cos \left (d x +c \right ) d^{2} e \,f^{2} x -12 \sin \left (d x +c \right )^{2} d^{3} e^{2} f x -12 \sin \left (d x +c \right )^{2} d^{3} e \,f^{2} x^{2}+24 \sin \left (d x +c \right ) d^{3} e^{2} f x +24 \sin \left (d x +c \right ) d^{3} e \,f^{2} x^{2}}{8 a \,d^{4}} \] Input:

int((f*x+e)^3*cos(d*x+c)^3/(a+a*sin(d*x+c)),x)
 

Output:

( - 6*cos(c + d*x)*sin(c + d*x)*d**2*e**2*f - 12*cos(c + d*x)*sin(c + d*x) 
*d**2*e*f**2*x - 6*cos(c + d*x)*sin(c + d*x)*d**2*f**3*x**2 + 3*cos(c + d* 
x)*sin(c + d*x)*f**3 + 24*cos(c + d*x)*d**2*e**2*f + 48*cos(c + d*x)*d**2* 
e*f**2*x + 24*cos(c + d*x)*d**2*f**3*x**2 - 48*cos(c + d*x)*f**3 - 4*sin(c 
 + d*x)**2*d**3*e**3 - 12*sin(c + d*x)**2*d**3*e**2*f*x - 12*sin(c + d*x)* 
*2*d**3*e*f**2*x**2 - 4*sin(c + d*x)**2*d**3*f**3*x**3 + 6*sin(c + d*x)**2 
*d*e*f**2 + 6*sin(c + d*x)**2*d*f**3*x + 8*sin(c + d*x)*d**3*e**3 + 24*sin 
(c + d*x)*d**3*e**2*f*x + 24*sin(c + d*x)*d**3*e*f**2*x**2 + 8*sin(c + d*x 
)*d**3*f**3*x**3 - 48*sin(c + d*x)*d*e*f**2 - 48*sin(c + d*x)*d*f**3*x + 8 
*d**3*e**3 + 6*d**3*e**2*f*x + 6*d**3*e*f**2*x**2 + 2*d**3*f**3*x**3 - 12* 
d*e*f**2 - 3*d*f**3*x)/(8*a*d**4)