\(\int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [317]

Optimal result

Integrand size = 27, antiderivative size = 61 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {x}{a^3}-\frac {7 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (a+a \sin (c+d x))^2} \] Output:

-x/a^3-7/3*cos(d*x+c)/a^3/d/(1+sin(d*x+c))+2/3*cos(d*x+c)/a/d/(a+a*sin(d*x 
+c))^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(61)=122\).

Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.38 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {180 d x \cos \left (\frac {d x}{2}\right )-351 \cos \left (c+\frac {d x}{2}\right )+277 \cos \left (c+\frac {3 d x}{2}\right )-60 d x \cos \left (2 c+\frac {3 d x}{2}\right )-471 \sin \left (\frac {d x}{2}\right )+180 d x \sin \left (c+\frac {d x}{2}\right )+60 d x \sin \left (c+\frac {3 d x}{2}\right )+3 \sin \left (2 c+\frac {3 d x}{2}\right )}{120 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \] Input:

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

Output:

-1/120*(180*d*x*Cos[(d*x)/2] - 351*Cos[c + (d*x)/2] + 277*Cos[c + (3*d*x)/ 
2] - 60*d*x*Cos[2*c + (3*d*x)/2] - 471*Sin[(d*x)/2] + 180*d*x*Sin[c + (d*x 
)/2] + 60*d*x*Sin[c + (3*d*x)/2] + 3*Sin[2*c + (3*d*x)/2])/(a^3*d*(Cos[c/2 
] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 3336, 25, 3042, 3214, 3042, 3127}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3336

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3214

\(\displaystyle \frac {7 a \int \frac {1}{\sin (c+d x) a+a}dx-3 x}{3 a^3}+\frac {2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 a \int \frac {1}{\sin (c+d x) a+a}dx-3 x}{3 a^3}+\frac {2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2}\)

\(\Big \downarrow \) 3127

\(\displaystyle \frac {-\frac {7 a \cos (c+d x)}{d (a \sin (c+d x)+a)}-3 x}{3 a^3}+\frac {2 \cos (c+d x)}{3 a d (a \sin (c+d x)+a)^2}\)

Input:

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

Output:

(2*Cos[c + d*x])/(3*a*d*(a + a*Sin[c + d*x])^2) + (-3*x - (7*a*Cos[c + d*x 
])/(d*(a + a*Sin[c + d*x])))/(3*a^3)
 

Defintions of rubi rules used

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 3127

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + 
 d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[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 3336

Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*( 
(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(b*c - a*d)*Cos 
[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3))), x] + Simp[1/(b^ 
3*(2*m + 3))   Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d* 
(2*m + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 
- b^2, 0] && LtQ[m, -3/2]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {x}{a^{3}}-\frac {2 \left (12 i {\mathrm e}^{i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}-7\right )}{3 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3}}\)	\(55\)
derivativedivides	\(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\)	\(67\)
default	\(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\)	\(67\)
parallelrisch	\(\frac {-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} x d +\left (-9 d x -6\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-9 d x -24\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 d x -10}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\)	\(77\)
norman	\(\frac {-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}-\frac {12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d a}-\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d a}-\frac {x}{a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {13 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {25 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a}-\frac {38 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {46 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a}-\frac {46 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {38 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a}-\frac {25 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}-\frac {13 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{a}-\frac {10}{3 a d}-\frac {76 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d a}-\frac {152 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d a}-\frac {44 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d a}-\frac {94 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d a}-\frac {220 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d a}-\frac {82 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(421\)

Input:

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

Output:

-x/a^3-2/3*(12*I*exp(I*(d*x+c))+9*exp(2*I*(d*x+c))-7)/d/a^3/(exp(I*(d*x+c) 
)+I)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (57) = 114\).

Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.03 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {{\left (3 \, d x - 7\right )} \cos \left (d x + c\right )^{2} - 6 \, d x - {\left (3 \, d x + 5\right )} \cos \left (d x + c\right ) - {\left (6 \, d x + {\left (3 \, d x + 7\right )} \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) + 2}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \] Input:

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

Output:

-1/3*((3*d*x - 7)*cos(d*x + c)^2 - 6*d*x - (3*d*x + 5)*cos(d*x + c) - (6*d 
*x + (3*d*x + 7)*cos(d*x + c) + 2)*sin(d*x + c) + 2)/(a^3*d*cos(d*x + c)^2 
 - a^3*d*cos(d*x + c) - 2*a^3*d - (a^3*d*cos(d*x + c) + 2*a^3*d)*sin(d*x + 
 c))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (53) = 106\).

Time = 8.44 (sec) , antiderivative size = 529, normalized size of antiderivative = 8.67 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\begin {cases} - \frac {3 d x \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {9 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {9 d x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {3 d x}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {6 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {24 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} - \frac {10}{3 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\left (c \right )} \cos ^{2}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-3*d*x*tan(c/2 + d*x/2)**3/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a* 
*3*d*tan(c/2 + d*x/2)**2 + 9*a**3*d*tan(c/2 + d*x/2) + 3*a**3*d) - 9*d*x*t 
an(c/2 + d*x/2)**2/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/ 
2)**2 + 9*a**3*d*tan(c/2 + d*x/2) + 3*a**3*d) - 9*d*x*tan(c/2 + d*x/2)/(3* 
a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/2)**2 + 9*a**3*d*tan(c 
/2 + d*x/2) + 3*a**3*d) - 3*d*x/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*t 
an(c/2 + d*x/2)**2 + 9*a**3*d*tan(c/2 + d*x/2) + 3*a**3*d) - 6*tan(c/2 + d 
*x/2)**2/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/2)**2 + 9* 
a**3*d*tan(c/2 + d*x/2) + 3*a**3*d) - 24*tan(c/2 + d*x/2)/(3*a**3*d*tan(c/ 
2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/2)**2 + 9*a**3*d*tan(c/2 + d*x/2) + 
 3*a**3*d) - 10/(3*a**3*d*tan(c/2 + d*x/2)**3 + 9*a**3*d*tan(c/2 + d*x/2)* 
*2 + 9*a**3*d*tan(c/2 + d*x/2) + 3*a**3*d), Ne(d, 0)), (x*sin(c)*cos(c)**2 
/(a*sin(c) + a)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (57) = 114\).

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (\frac {\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{3 \, d} \] Input:

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

Output:

-2/3*((12*sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) 
 + 1)^2 + 5)/(a^3 + 3*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 3*a^3*sin(d*x 
+ c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) + 3 
*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \] Input:

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

Output:

-1/3*(3*(d*x + c)/a^3 + 2*(3*tan(1/2*d*x + 1/2*c)^2 + 12*tan(1/2*d*x + 1/2 
*c) + 5)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^3))/d
 

Mupad [B] (verification not implemented)

Time = 17.73 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {x}{a^3}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {10}{3}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \] Input:

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

Output:

- x/a^3 - (8*tan(c/2 + (d*x)/2) + 2*tan(c/2 + (d*x)/2)^2 + 10/3)/(a^3*d*(t 
an(c/2 + (d*x)/2) + 1)^3)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 645, normalized size of antiderivative = 10.57 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*cos(c + d*x)*sin(c + d*x)*tan((c + d*x)/2)**3 - 18*cos(c + d*x)*sin( 
c + d*x)*tan((c + d*x)/2)**2 - 18*cos(c + d*x)*sin(c + d*x)*tan((c + d*x)/ 
2) - 6*cos(c + d*x)*sin(c + d*x) - 3*cos(c + d*x)*tan((c + d*x)/2)**3 - 9* 
cos(c + d*x)*tan((c + d*x)/2)**2 - 9*cos(c + d*x)*tan((c + d*x)/2) - 3*cos 
(c + d*x) - 6*sin(c + d*x)**2*tan((c + d*x)/2)**3*d*x + 4*sin(c + d*x)**2* 
tan((c + d*x)/2)**3 - 18*sin(c + d*x)**2*tan((c + d*x)/2)**2*d*x - 18*sin( 
c + d*x)**2*tan((c + d*x)/2)*d*x - 18*sin(c + d*x)**2*tan((c + d*x)/2) - 6 
*sin(c + d*x)**2*d*x - 10*sin(c + d*x)**2 - 12*sin(c + d*x)*tan((c + d*x)/ 
2)**3*d*x + 8*sin(c + d*x)*tan((c + d*x)/2)**3 - 36*sin(c + d*x)*tan((c + 
d*x)/2)**2*d*x - 36*sin(c + d*x)*tan((c + d*x)/2)*d*x - 36*sin(c + d*x)*ta 
n((c + d*x)/2) - 12*sin(c + d*x)*d*x - 20*sin(c + d*x) - 6*tan((c + d*x)/2 
)**3*d*x + 4*tan((c + d*x)/2)**3 - 18*tan((c + d*x)/2)**2*d*x - 18*tan((c 
+ d*x)/2)*d*x - 18*tan((c + d*x)/2) - 6*d*x - 10)/(6*a**3*d*(sin(c + d*x)* 
*2*tan((c + d*x)/2)**3 + 3*sin(c + d*x)**2*tan((c + d*x)/2)**2 + 3*sin(c + 
 d*x)**2*tan((c + d*x)/2) + sin(c + d*x)**2 + 2*sin(c + d*x)*tan((c + d*x) 
/2)**3 + 6*sin(c + d*x)*tan((c + d*x)/2)**2 + 6*sin(c + d*x)*tan((c + d*x) 
/2) + 2*sin(c + d*x) + tan((c + d*x)/2)**3 + 3*tan((c + d*x)/2)**2 + 3*tan 
((c + d*x)/2) + 1))