Integrand size = 29, antiderivative size = 87 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {11 x}{2 a^3}-\frac {5 \cos (c+d x)}{a^3 d}+\frac {\cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))} \]
-11/2*x/a^3-5*cos(d*x+c)/a^3/d+1/3*cos(d*x+c)^3/a^3/d+3/2*cos(d*x+c)*sin(d *x+c)/a^3/d-4*cos(d*x+c)/a^3/d/(1+sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(87)=174\).
Time = 1.41 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.08 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {(1-660 d x) \cos \left (\frac {d x}{2}\right )-286 \cos \left (c+\frac {d x}{2}\right )-240 \cos \left (c+\frac {3 d x}{2}\right )-40 \cos \left (3 c+\frac {5 d x}{2}\right )+5 \cos \left (3 c+\frac {7 d x}{2}\right )+1244 \sin \left (\frac {d x}{2}\right )+\sin \left (c+\frac {d x}{2}\right )-660 d x \sin \left (c+\frac {d x}{2}\right )-240 \sin \left (2 c+\frac {3 d x}{2}\right )+40 \sin \left (2 c+\frac {5 d x}{2}\right )+5 \sin \left (4 c+\frac {7 d x}{2}\right )}{120 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
((1 - 660*d*x)*Cos[(d*x)/2] - 286*Cos[c + (d*x)/2] - 240*Cos[c + (3*d*x)/2 ] - 40*Cos[3*c + (5*d*x)/2] + 5*Cos[3*c + (7*d*x)/2] + 1244*Sin[(d*x)/2] + Sin[c + (d*x)/2] - 660*d*x*Sin[c + (d*x)/2] - 240*Sin[2*c + (3*d*x)/2] + 40*Sin[2*c + (5*d*x)/2] + 5*Sin[4*c + (7*d*x)/2])/(120*a^3*d*(Cos[c/2] + S in[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.45 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3354, 3042, 3188, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^4(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^4}{(a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int (a-a \sin (c+d x))^3 \tan ^2(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (a-a \sin (c+d x))^3 \tan (c+d x)^2dx}{a^6}\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle \frac {\int \left (a \sin ^3(c+d x)-3 a \sin ^2(c+d x)+4 a \sin (c+d x)-4 a+\frac {4 a}{\sin (c+d x)+1}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a \cos ^3(c+d x)}{3 d}-\frac {5 a \cos (c+d x)}{d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{2 d}-\frac {4 a \cos (c+d x)}{d (\sin (c+d x)+1)}-\frac {11 a x}{2}}{a^4}\) |
((-11*a*x)/2 - (5*a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (3*a*Cos[ c + d*x]*Sin[c + d*x])/(2*d) - (4*a*Cos[c + d*x])/(d*(1 + Sin[c + d*x])))/ a^4
3.5.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(-\frac {11 x}{2 a^{3}}-\frac {19 \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {19 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {8}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {\cos \left (3 d x +3 c \right )}{12 d \,a^{3}}+\frac {3 \sin \left (2 d x +2 c \right )}{4 d \,a^{3}}\) | \(98\) |
| derivativedivides | \(\frac {-\frac {8 \left (\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {7}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(104\) |
| default | \(\frac {-\frac {8 \left (\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {7}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(104\) |
| parallelrisch | \(\frac {-132 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-132 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+8 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-48 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-48 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-8 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+209 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-97 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{3} \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(141\) |
| norman | \(\frac {-\frac {748 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {220 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1408 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {88 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1100 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {440 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1595 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1595 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1408 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1100 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {52}{3 a d}-\frac {748 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {440 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {11 x}{2 a}-\frac {4105 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {667 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {220 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {88 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {55 x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {423 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {227 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {11 x \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {55 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {7487 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {55 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {7225 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {1555 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {7855 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {4591 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {6295 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {2891 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {175 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {5825 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {827 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {11 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(637\) |
-11/2*x/a^3-19/8/d/a^3*exp(I*(d*x+c))-19/8/d/a^3*exp(-I*(d*x+c))-8/d/a^3/( exp(I*(d*x+c))+I)+1/12/d/a^3*cos(3*d*x+3*c)+3/4/d/a^3*sin(2*d*x+2*c)
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{3} - 33 \, d x - 3 \, {\left (11 \, d x + 15\right )} \cos \left (d x + c\right ) - 30 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right )^{3} - 33 \, d x + 9 \, \cos \left (d x + c\right )^{2} - 21 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right ) - 24}{6 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
1/6*(2*cos(d*x + c)^4 - 7*cos(d*x + c)^3 - 33*d*x - 3*(11*d*x + 15)*cos(d* x + c) - 30*cos(d*x + c)^2 + (2*cos(d*x + c)^3 - 33*d*x + 9*cos(d*x + c)^2 - 21*cos(d*x + c) + 24)*sin(d*x + c) - 24)/(a^3*d*cos(d*x + c) + a^3*d*si n(d*x + c) + a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 2264 vs. \(2 (80) = 160\).
Time = 37.43 (sec) , antiderivative size = 2264, normalized size of antiderivative = 26.02 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
Piecewise((-33*d*x*tan(c/2 + d*x/2)**7/(6*a**3*d*tan(c/2 + d*x/2)**7 + 6*a **3*d*tan(c/2 + d*x/2)**6 + 18*a**3*d*tan(c/2 + d*x/2)**5 + 18*a**3*d*tan( c/2 + d*x/2)**4 + 18*a**3*d*tan(c/2 + d*x/2)**3 + 18*a**3*d*tan(c/2 + d*x/ 2)**2 + 6*a**3*d*tan(c/2 + d*x/2) + 6*a**3*d) - 33*d*x*tan(c/2 + d*x/2)**6 /(6*a**3*d*tan(c/2 + d*x/2)**7 + 6*a**3*d*tan(c/2 + d*x/2)**6 + 18*a**3*d* tan(c/2 + d*x/2)**5 + 18*a**3*d*tan(c/2 + d*x/2)**4 + 18*a**3*d*tan(c/2 + d*x/2)**3 + 18*a**3*d*tan(c/2 + d*x/2)**2 + 6*a**3*d*tan(c/2 + d*x/2) + 6* a**3*d) - 99*d*x*tan(c/2 + d*x/2)**5/(6*a**3*d*tan(c/2 + d*x/2)**7 + 6*a** 3*d*tan(c/2 + d*x/2)**6 + 18*a**3*d*tan(c/2 + d*x/2)**5 + 18*a**3*d*tan(c/ 2 + d*x/2)**4 + 18*a**3*d*tan(c/2 + d*x/2)**3 + 18*a**3*d*tan(c/2 + d*x/2) **2 + 6*a**3*d*tan(c/2 + d*x/2) + 6*a**3*d) - 99*d*x*tan(c/2 + d*x/2)**4/( 6*a**3*d*tan(c/2 + d*x/2)**7 + 6*a**3*d*tan(c/2 + d*x/2)**6 + 18*a**3*d*ta n(c/2 + d*x/2)**5 + 18*a**3*d*tan(c/2 + d*x/2)**4 + 18*a**3*d*tan(c/2 + d* x/2)**3 + 18*a**3*d*tan(c/2 + d*x/2)**2 + 6*a**3*d*tan(c/2 + d*x/2) + 6*a* *3*d) - 99*d*x*tan(c/2 + d*x/2)**3/(6*a**3*d*tan(c/2 + d*x/2)**7 + 6*a**3* d*tan(c/2 + d*x/2)**6 + 18*a**3*d*tan(c/2 + d*x/2)**5 + 18*a**3*d*tan(c/2 + d*x/2)**4 + 18*a**3*d*tan(c/2 + d*x/2)**3 + 18*a**3*d*tan(c/2 + d*x/2)** 2 + 6*a**3*d*tan(c/2 + d*x/2) + 6*a**3*d) - 99*d*x*tan(c/2 + d*x/2)**2/(6* a**3*d*tan(c/2 + d*x/2)**7 + 6*a**3*d*tan(c/2 + d*x/2)**6 + 18*a**3*d*tan( c/2 + d*x/2)**5 + 18*a**3*d*tan(c/2 + d*x/2)**4 + 18*a**3*d*tan(c/2 + d...
Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (81) = 162\).
Time = 0.32 (sec) , antiderivative size = 312, normalized size of antiderivative = 3.59 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {19 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {123 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {60 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {96 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {33 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {33 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 52}{a^{3} + \frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {33 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
-1/3*((19*sin(d*x + c)/(cos(d*x + c) + 1) + 123*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 60*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 96*sin(d*x + c)^4/(co s(d*x + c) + 1)^4 + 33*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 33*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 52)/(a^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 + 3*a^3*sin(d*x + c)^3/(cos(d *x + c) + 1)^3 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a ^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7) + 33*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {33 \, {\left (d x + c\right )}}{a^{3}} + \frac {48}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 28\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}}{6 \, d} \]
-1/6*(33*(d*x + c)/a^3 + 48/(a^3*(tan(1/2*d*x + 1/2*c) + 1)) + 2*(9*tan(1/ 2*d*x + 1/2*c)^5 + 24*tan(1/2*d*x + 1/2*c)^4 + 60*tan(1/2*d*x + 1/2*c)^2 - 9*tan(1/2*d*x + 1/2*c) + 28)/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^3))/d
Time = 13.64 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.39 \[ \int \frac {\cos ^4(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {11\,x}{2\,a^3}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {19\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {52}{3}}{a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]