Integrand size = 21, antiderivative size = 104 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {7 x}{16 a^2}+\frac {7 \cos ^5(c+d x)}{30 a^2 d}+\frac {7 \cos (c+d x) \sin (c+d x)}{16 a^2 d}+\frac {7 \cos ^3(c+d x) \sin (c+d x)}{24 a^2 d}+\frac {\cos ^7(c+d x)}{6 d \left (a^2+a^2 \sin (c+d x)\right )} \]
7/16*x/a^2+7/30*cos(d*x+c)^5/a^2/d+7/16*cos(d*x+c)*sin(d*x+c)/a^2/d+7/24*c os(d*x+c)^3*sin(d*x+c)/a^2/d+1/6*cos(d*x+c)^7/d/(a^2+a^2*sin(d*x+c))
Time = 0.76 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\cos ^9(c+d x) \left (-210 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (96+39 \sin (c+d x)-327 \sin ^2(c+d x)+202 \sin ^3(c+d x)+86 \sin ^4(c+d x)-136 \sin ^5(c+d x)+40 \sin ^6(c+d x)\right )\right )}{240 a^2 d (-1+\sin (c+d x))^5 (1+\sin (c+d x))^{9/2}} \]
-1/240*(Cos[c + d*x]^9*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(96 + 39*Sin[c + d*x] - 327*Sin[ c + d*x]^2 + 202*Sin[c + d*x]^3 + 86*Sin[c + d*x]^4 - 136*Sin[c + d*x]^5 + 40*Sin[c + d*x]^6)))/(a^2*d*(-1 + Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2 ))
Time = 0.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3158, 3042, 3161, 3042, 3115, 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.
| \(\displaystyle \int \frac {\cos ^8(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle \frac {7 \int \frac {\cos ^6(c+d x)}{\sin (c+d x) a+a}dx}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \int \frac {\cos (c+d x)^6}{\sin (c+d x) a+a}dx}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle \frac {7 \left (\frac {\int \cos ^4(c+d x)dx}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {7 \left (\frac {\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {7 \left (\frac {\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}}{a}+\frac {\cos ^5(c+d x)}{5 a d}\right )}{6 a}+\frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\cos ^7(c+d x)}{6 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {7 \left (\frac {\cos ^5(c+d x)}{5 a d}+\frac {\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a}\right )}{6 a}\) |
Cos[c + d*x]^7/(6*d*(a^2 + a^2*Sin[c + d*x])) + (7*(Cos[c + d*x]^5/(5*a*d) + ((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4)/a))/(6*a)
3.1.62.3.1 Defintions of rubi rules used
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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(a*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && In tegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Time = 0.49 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.75
| method | result | size |
| parallelrisch | \(\frac {420 d x +240 \cos \left (d x +c \right )-5 \sin \left (6 d x +6 c \right )+24 \cos \left (5 d x +5 c \right )+15 \sin \left (4 d x +4 c \right )+120 \cos \left (3 d x +3 c \right )+255 \sin \left (2 d x +2 c \right )+384}{960 a^{2} d}\) | \(78\) |
| risch | \(\frac {7 x}{16 a^{2}}+\frac {\cos \left (d x +c \right )}{4 a^{2} d}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}+\frac {\cos \left (5 d x +5 c \right )}{40 a^{2} d}+\frac {\sin \left (4 d x +4 c \right )}{64 a^{2} d}+\frac {\cos \left (3 d x +3 c \right )}{8 a^{2} d}+\frac {17 \sin \left (2 d x +2 c \right )}{64 a^{2} d}\) | \(107\) |
| derivativedivides | \(\frac {\frac {2 \left (-\frac {9 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {89 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {89 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}+\frac {2}{5}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\) | \(181\) |
| default | \(\frac {\frac {2 \left (-\frac {9 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+2 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {89 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+2 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {89 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}+\frac {2}{5}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{2} d}\) | \(181\) |
1/960*(420*d*x+240*cos(d*x+c)-5*sin(6*d*x+6*c)+24*cos(5*d*x+5*c)+15*sin(4* d*x+4*c)+120*cos(3*d*x+3*c)+255*sin(2*d*x+2*c)+384)/a^2/d
Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {96 \, \cos \left (d x + c\right )^{5} + 105 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 14 \, \cos \left (d x + c\right )^{3} - 21 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, a^{2} d} \]
1/240*(96*cos(d*x + c)^5 + 105*d*x - 5*(8*cos(d*x + c)^5 - 14*cos(d*x + c) ^3 - 21*cos(d*x + c))*sin(d*x + c))/(a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 2531 vs. \(2 (95) = 190\).
Time = 56.72 (sec) , antiderivative size = 2531, normalized size of antiderivative = 24.34 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((105*d*x*tan(c/2 + d*x/2)**12/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800 *a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2* d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 630*d*x*tan(c/2 + d*x/2)**10/(240*a* *2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d *tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c /2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 1575*d*x* tan(c/2 + d*x/2)**8/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d* x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)** 2 + 240*a**2*d) + 2100*d*x*tan(c/2 + d*x/2)**6/(240*a**2*d*tan(c/2 + d*x/2 )**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 144 0*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 1575*d*x*tan(c/2 + d*x/2)**4/ (240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600 *a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2* d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 63 0*d*x*tan(c/2 + d*x/2)**2/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*t an(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/ 2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 +...
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.78 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {135 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {96 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {445 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {960 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {330 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {960 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {480 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {445 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {480 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {135 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 96}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \]
1/120*((135*sin(d*x + c)/(cos(d*x + c) + 1) + 96*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 445*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 960*sin(d*x + c)^4/ (cos(d*x + c) + 1)^4 - 330*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 960*sin(d *x + c)^6/(cos(d*x + c) + 1)^6 + 330*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 480*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 445*sin(d*x + c)^9/(cos(d*x + c ) + 1)^9 + 480*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 135*sin(d*x + c)^11 /(cos(d*x + c) + 1)^11 + 96)/(a^2 + 6*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1 )^2 + 15*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a^2*sin(d*x + c)^6/( cos(d*x + c) + 1)^6 + 15*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a^2*s in(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) + 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.72 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {105 \, {\left (d x + c\right )}}{a^{2}} - \frac {2 \, {\left (135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 445 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 135 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \]
1/240*(105*(d*x + c)/a^2 - 2*(135*tan(1/2*d*x + 1/2*c)^11 - 480*tan(1/2*d* x + 1/2*c)^10 + 445*tan(1/2*d*x + 1/2*c)^9 - 480*tan(1/2*d*x + 1/2*c)^8 - 330*tan(1/2*d*x + 1/2*c)^7 - 960*tan(1/2*d*x + 1/2*c)^6 + 330*tan(1/2*d*x + 1/2*c)^5 - 960*tan(1/2*d*x + 1/2*c)^4 - 445*tan(1/2*d*x + 1/2*c)^3 - 96* tan(1/2*d*x + 1/2*c)^2 - 135*tan(1/2*d*x + 1/2*c) - 96)/((tan(1/2*d*x + 1/ 2*c)^2 + 1)^6*a^2))/d
Time = 10.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.65 \[ \int \frac {\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {7\,x}{16\,a^2}+\frac {-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {89\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {4}{5}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
(7*x)/(16*a^2) + ((9*tan(c/2 + (d*x)/2))/8 + (4*tan(c/2 + (d*x)/2)^2)/5 + (89*tan(c/2 + (d*x)/2)^3)/24 + 8*tan(c/2 + (d*x)/2)^4 - (11*tan(c/2 + (d*x )/2)^5)/4 + 8*tan(c/2 + (d*x)/2)^6 + (11*tan(c/2 + (d*x)/2)^7)/4 + 4*tan(c /2 + (d*x)/2)^8 - (89*tan(c/2 + (d*x)/2)^9)/24 + 4*tan(c/2 + (d*x)/2)^10 - (9*tan(c/2 + (d*x)/2)^11)/8 + 4/5)/(a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^6)