\(\int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [645]

Optimal result

Integrand size = 29, antiderivative size = 105 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13 x}{8 a^3}+\frac {4 \cos (c+d x)}{a^3 d}-\frac {5 \cos ^3(c+d x)}{3 a^3 d}+\frac {\cos ^5(c+d x)}{5 a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {3 \cos (c+d x) \sin ^3(c+d x)}{4 a^3 d} \] Output:

13/8*x/a^3+4*cos(d*x+c)/a^3/d-5/3*cos(d*x+c)^3/a^3/d+1/5*cos(d*x+c)^5/a^3/ 
d-13/8*cos(d*x+c)*sin(d*x+c)/a^3/d-3/4*cos(d*x+c)*sin(d*x+c)^3/a^3/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(310\) vs. \(2(105)=210\).

Time = 2.23 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.95 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {1560 d x \cos \left (\frac {c}{2}\right )+1380 \cos \left (\frac {c}{2}+d x\right )+1380 \cos \left (\frac {3 c}{2}+d x\right )-480 \cos \left (\frac {3 c}{2}+2 d x\right )+480 \cos \left (\frac {5 c}{2}+2 d x\right )-170 \cos \left (\frac {5 c}{2}+3 d x\right )-170 \cos \left (\frac {7 c}{2}+3 d x\right )+45 \cos \left (\frac {7 c}{2}+4 d x\right )-45 \cos \left (\frac {9 c}{2}+4 d x\right )+6 \cos \left (\frac {9 c}{2}+5 d x\right )+6 \cos \left (\frac {11 c}{2}+5 d x\right )+10 \sin \left (\frac {c}{2}\right )+1560 d x \sin \left (\frac {c}{2}\right )-1380 \sin \left (\frac {c}{2}+d x\right )+1380 \sin \left (\frac {3 c}{2}+d x\right )-480 \sin \left (\frac {3 c}{2}+2 d x\right )-480 \sin \left (\frac {5 c}{2}+2 d x\right )+170 \sin \left (\frac {5 c}{2}+3 d x\right )-170 \sin \left (\frac {7 c}{2}+3 d x\right )+45 \sin \left (\frac {7 c}{2}+4 d x\right )+45 \sin \left (\frac {9 c}{2}+4 d x\right )-6 \sin \left (\frac {9 c}{2}+5 d x\right )+6 \sin \left (\frac {11 c}{2}+5 d x\right )}{960 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \] Input:

Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
 

Output:

(1560*d*x*Cos[c/2] + 1380*Cos[c/2 + d*x] + 1380*Cos[(3*c)/2 + d*x] - 480*C 
os[(3*c)/2 + 2*d*x] + 480*Cos[(5*c)/2 + 2*d*x] - 170*Cos[(5*c)/2 + 3*d*x] 
- 170*Cos[(7*c)/2 + 3*d*x] + 45*Cos[(7*c)/2 + 4*d*x] - 45*Cos[(9*c)/2 + 4* 
d*x] + 6*Cos[(9*c)/2 + 5*d*x] + 6*Cos[(11*c)/2 + 5*d*x] + 10*Sin[c/2] + 15 
60*d*x*Sin[c/2] - 1380*Sin[c/2 + d*x] + 1380*Sin[(3*c)/2 + d*x] - 480*Sin[ 
(3*c)/2 + 2*d*x] - 480*Sin[(5*c)/2 + 2*d*x] + 170*Sin[(5*c)/2 + 3*d*x] - 1 
70*Sin[(7*c)/2 + 3*d*x] + 45*Sin[(7*c)/2 + 4*d*x] + 45*Sin[(9*c)/2 + 4*d*x 
] - 6*Sin[(9*c)/2 + 5*d*x] + 6*Sin[(11*c)/2 + 5*d*x])/(960*a^3*d*(Cos[c/2] 
 + Sin[c/2]))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3348, 3042, 3236, 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.

\(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^6(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3348

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3236

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {5 a^3 \cos ^3(c+d x)}{3 d}+\frac {4 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {13 a^3 x}{8}}{a^6}\)

Input:

Int[(Cos[c + d*x]^6*Sin[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]
 

Output:

((13*a^3*x)/8 + (4*a^3*Cos[c + d*x])/d - (5*a^3*Cos[c + d*x]^3)/(3*d) + (a 
^3*Cos[c + d*x]^5)/(5*d) - (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (3*a 
^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d))/a^6
 

Defintions of rubi rules used

rule 2009

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

rule 3042

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

rule 3236

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + 
f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt 
Q[m, 0] && RationalQ[n]
 

rule 3348

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

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {780 d x +6 \cos \left (5 d x +5 c \right )-170 \cos \left (3 d x +3 c \right )+1380 \cos \left (d x +c \right )+45 \sin \left (4 d x +4 c \right )-480 \sin \left (2 d x +2 c \right )+1216}{480 d \,a^{3}}\)	\(67\)
risch	\(\frac {13 x}{8 a^{3}}+\frac {23 \cos \left (d x +c \right )}{8 a^{3} d}+\frac {\cos \left (5 d x +5 c \right )}{80 a^{3} d}+\frac {3 \sin \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {17 \cos \left (3 d x +3 c \right )}{48 a^{3} d}-\frac {\sin \left (2 d x +2 c \right )}{d \,a^{3}}\)	\(90\)
derivativedivides	\(\frac {\frac {8 \left (\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{32}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{16}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{6}-\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{16}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{6}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {19}{30}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{3}}\)	\(129\)
default	\(\frac {\frac {8 \left (\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{32}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{16}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{6}-\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{16}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{6}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\frac {19}{30}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{3}}\)	\(129\)

Input:

int(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/480*(780*d*x+6*cos(5*d*x+5*c)-170*cos(3*d*x+3*c)+1380*cos(d*x+c)+45*sin( 
4*d*x+4*c)-480*sin(2*d*x+2*c)+1216)/d/a^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {24 \, \cos \left (d x + c\right )^{5} - 200 \, \cos \left (d x + c\right )^{3} + 195 \, d x + 15 \, {\left (6 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{120 \, a^{3} d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="frica 
s")
 

Output:

1/120*(24*cos(d*x + c)^5 - 200*cos(d*x + c)^3 + 195*d*x + 15*(6*cos(d*x + 
c)^3 - 19*cos(d*x + c))*sin(d*x + c) + 480*cos(d*x + c))/(a^3*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (99) = 198\).

Time = 92.53 (sec) , antiderivative size = 1608, normalized size of antiderivative = 15.31 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)**6*sin(d*x+c)**2/(a+a*sin(d*x+c))**3,x)
 

Output:

Piecewise((195*d*x*tan(c/2 + d*x/2)**10/(120*a**3*d*tan(c/2 + d*x/2)**10 + 
 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a 
**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 
 975*d*x*tan(c/2 + d*x/2)**8/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d 
*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c 
/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 1950*d*x*t 
an(c/2 + d*x/2)**6/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + 
 d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2 
)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 1950*d*x*tan(c/2 + d 
*x/2)**4/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 
 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600 
*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 975*d*x*tan(c/2 + d*x/2)**2/(1 
20*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a** 
3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan 
(c/2 + d*x/2)**2 + 120*a**3*d) + 195*d*x/(120*a**3*d*tan(c/2 + d*x/2)**10 
+ 600*a**3*d*tan(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200* 
a**3*d*tan(c/2 + d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) 
+ 390*tan(c/2 + d*x/2)**9/(120*a**3*d*tan(c/2 + d*x/2)**10 + 600*a**3*d*ta 
n(c/2 + d*x/2)**8 + 1200*a**3*d*tan(c/2 + d*x/2)**6 + 1200*a**3*d*tan(c/2 
+ d*x/2)**4 + 600*a**3*d*tan(c/2 + d*x/2)**2 + 120*a**3*d) + 1500*tan(c...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (95) = 190\).

Time = 0.13 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.76 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {195 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1520 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2320 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {750 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {195 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 304}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {195 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="maxim 
a")
 

Output:

-1/60*((195*sin(d*x + c)/(cos(d*x + c) + 1) - 1520*sin(d*x + c)^2/(cos(d*x 
 + c) + 1)^2 + 750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 2320*sin(d*x + c) 
^4/(cos(d*x + c) + 1)^4 - 720*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 750*si 
n(d*x + c)^7/(cos(d*x + c) + 1)^7 - 195*sin(d*x + c)^9/(cos(d*x + c) + 1)^ 
9 - 304)/(a^3 + 5*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^3*sin(d*x 
 + c)^4/(cos(d*x + c) + 1)^4 + 10*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 
+ 5*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a^3*sin(d*x + c)^10/(cos(d*x 
 + c) + 1)^10) - 195*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.21 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {195 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 304\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a^{3}}}{120 \, d} \] Input:

integrate(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^3,x, algorithm="giac" 
)
 

Output:

1/120*(195*(d*x + c)/a^3 + 2*(195*tan(1/2*d*x + 1/2*c)^9 + 750*tan(1/2*d*x 
 + 1/2*c)^7 + 720*tan(1/2*d*x + 1/2*c)^6 + 2320*tan(1/2*d*x + 1/2*c)^4 - 7 
50*tan(1/2*d*x + 1/2*c)^3 + 1520*tan(1/2*d*x + 1/2*c)^2 - 195*tan(1/2*d*x 
+ 1/2*c) + 304)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*a^3))/d
 

Mupad [B] (verification not implemented)

Time = 33.94 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13\,x}{8\,a^3}+\frac {4\,\cos \left (c+d\,x\right )}{a^3\,d}-\frac {5\,{\cos \left (c+d\,x\right )}^3}{3\,a^3\,d}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3\,d}+\frac {3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,a^3\,d}-\frac {19\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,a^3\,d} \] Input:

int((cos(c + d*x)^6*sin(c + d*x)^2)/(a + a*sin(c + d*x))^3,x)
 

Output:

(13*x)/(8*a^3) + (4*cos(c + d*x))/(a^3*d) - (5*cos(c + d*x)^3)/(3*a^3*d) + 
 cos(c + d*x)^5/(5*a^3*d) + (3*cos(c + d*x)^3*sin(c + d*x))/(4*a^3*d) - (1 
9*cos(c + d*x)*sin(c + d*x))/(8*a^3*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-90 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+152 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-195 \cos \left (d x +c \right ) \sin \left (d x +c \right )+304 \cos \left (d x +c \right )+195 d x -304}{120 a^{3} d} \] Input:

int(cos(d*x+c)^6*sin(d*x+c)^2/(a+a*sin(d*x+c))^3,x)
 

Output:

(24*cos(c + d*x)*sin(c + d*x)**4 - 90*cos(c + d*x)*sin(c + d*x)**3 + 152*c 
os(c + d*x)*sin(c + d*x)**2 - 195*cos(c + d*x)*sin(c + d*x) + 304*cos(c + 
d*x) + 195*d*x - 304)/(120*a**3*d)