Integrand size = 29, antiderivative size = 104 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 x}{8 a}+\frac {\cos (c+d x)}{a d}-\frac {2 \cos ^3(c+d x)}{3 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{4 a d} \]
3/8*x/a+cos(d*x+c)/a/d-2/3*cos(d*x+c)^3/a/d+1/5*cos(d*x+c)^5/a/d-3/8*cos(d *x+c)*sin(d*x+c)/a/d-1/4*cos(d*x+c)*sin(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(281\) vs. \(2(104)=208\).
Time = 3.97 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1}{480} \left (\frac {180 x}{a}+\frac {300 \cos (c) \cos (d x)}{a d}-\frac {50 \cos (3 c) \cos (3 d x)}{a d}+\frac {6 \cos (5 c) \cos (5 d x)}{a d}-\frac {120 \cos (2 d x) \sin (2 c)}{a d}+\frac {15 \cos (4 d x) \sin (4 c)}{a d}-\frac {300 \sin (c) \sin (d x)}{a d}-\frac {120 \cos (2 c) \sin (2 d x)}{a d}+\frac {50 \sin (3 c) \sin (3 d x)}{a d}+\frac {15 \cos (4 c) \sin (4 d x)}{a d}-\frac {6 \sin (5 c) \sin (5 d x)}{a d}-\frac {60 \sin \left (\frac {d x}{2}\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {30 \sin (c+d x)}{a d (1+\sin (c+d x))}+\frac {60 \sin ^2\left (\frac {1}{2} (c+d x)\right )}{d (a+a \sin (c+d x))}\right ) \]
((180*x)/a + (300*Cos[c]*Cos[d*x])/(a*d) - (50*Cos[3*c]*Cos[3*d*x])/(a*d) + (6*Cos[5*c]*Cos[5*d*x])/(a*d) - (120*Cos[2*d*x]*Sin[2*c])/(a*d) + (15*Co s[4*d*x]*Sin[4*c])/(a*d) - (300*Sin[c]*Sin[d*x])/(a*d) - (120*Cos[2*c]*Sin [2*d*x])/(a*d) + (50*Sin[3*c]*Sin[3*d*x])/(a*d) + (15*Cos[4*c]*Sin[4*d*x]) /(a*d) - (6*Sin[5*c]*Sin[5*d*x])/(a*d) - (60*Sin[(d*x)/2])/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (30*Sin[c + d*x])/(a* d*(1 + Sin[c + d*x])) + (60*Sin[(c + d*x)/2]^2)/(d*(a + a*Sin[c + d*x])))/ 480
Time = 0.44 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3042, 3318, 3042, 3113, 2009, 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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos ^2(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \cos (c+d x)^2}{a \sin (c+d x)+a}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \sin ^4(c+d x)dx}{a}-\frac {\int \sin ^5(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin (c+d x)^4dx}{a}-\frac {\int \sin (c+d x)^5dx}{a}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\int \sin (c+d x)^4dx}{a}+\frac {\int \left (\cos ^4(c+d x)-2 \cos ^2(c+d x)+1\right )d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \sin (c+d x)^4dx}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{4} \int \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \int \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}}{a}+\frac {\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \cos ^5(c+d x)-\frac {2}{3} \cos ^3(c+d x)+\cos (c+d x)}{a d}+\frac {\frac {3}{4} \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}}{a}\) |
(Cos[c + d*x] - (2*Cos[c + d*x]^3)/3 + Cos[c + d*x]^5/5)/(a*d) + (-1/4*(Co s[c + d*x]*Sin[c + d*x]^3)/d + (3*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d) ))/4)/a
3.3.97.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64
| method | result | size |
| parallelrisch | \(\frac {180 d x +300 \cos \left (d x +c \right )+6 \cos \left (5 d x +5 c \right )+15 \sin \left (4 d x +4 c \right )-50 \cos \left (3 d x +3 c \right )-120 \sin \left (2 d x +2 c \right )+256}{480 d a}\) | \(67\) |
| risch | \(\frac {3 x}{8 a}+\frac {5 \cos \left (d x +c \right )}{8 a d}+\frac {\cos \left (5 d x +5 c \right )}{80 a d}+\frac {\sin \left (4 d x +4 c \right )}{32 d a}-\frac {5 \cos \left (3 d x +3 c \right )}{48 a d}-\frac {\sin \left (2 d x +2 c \right )}{4 d a}\) | \(90\) |
| derivativedivides | \(\frac {\frac {32 \left (\frac {3 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}+\frac {1}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(116\) |
| default | \(\frac {\frac {32 \left (\frac {3 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}+\frac {1}{30}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(116\) |
| norman | \(\frac {\frac {45 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {9 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {45 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {45 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {45 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {9 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {19}{60 a d}+\frac {3 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {3 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {3 x}{8 a}-\frac {15 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {23 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {3 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{30 d a}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {47 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d a}-\frac {31 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {47 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(490\) |
1/480*(180*d*x+300*cos(d*x+c)+6*cos(5*d*x+5*c)+15*sin(4*d*x+4*c)-50*cos(3* d*x+3*c)-120*sin(2*d*x+2*c)+256)/d/a
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {24 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} + 45 \, d x + 15 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 120 \, \cos \left (d x + c\right )}{120 \, a d} \]
1/120*(24*cos(d*x + c)^5 - 80*cos(d*x + c)^3 + 45*d*x + 15*(2*cos(d*x + c) ^3 - 5*cos(d*x + c))*sin(d*x + c) + 120*cos(d*x + c))/(a*d)
Leaf count of result is larger than twice the leaf count of optimal. 1360 vs. \(2 (85) = 170\).
Time = 11.69 (sec) , antiderivative size = 1360, normalized size of antiderivative = 13.08 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
Piecewise((45*d*x*tan(c/2 + d*x/2)**10/(120*a*d*tan(c/2 + d*x/2)**10 + 600 *a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 225*d*x*tan(c/2 + d*x/2)**8/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 12 00*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/ 2 + d*x/2)**2 + 120*a*d) + 450*d*x*tan(c/2 + d*x/2)**6/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 45 0*d*x*tan(c/2 + d*x/2)**4/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 225*d*x*tan(c/2 + d*x/2)**2/(12 0*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/ 2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 45*d*x/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x /2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600 *a*d*tan(c/2 + d*x/2)**2 + 120*a*d) + 90*tan(c/2 + d*x/2)**9/(120*a*d*tan( c/2 + d*x/2)**10 + 600*a*d*tan(c/2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2) **6 + 1200*a*d*tan(c/2 + d*x/2)**4 + 600*a*d*tan(c/2 + d*x/2)**2 + 120*a*d ) + 420*tan(c/2 + d*x/2)**7/(120*a*d*tan(c/2 + d*x/2)**10 + 600*a*d*tan(c/ 2 + d*x/2)**8 + 1200*a*d*tan(c/2 + d*x/2)**6 + 1200*a*d*tan(c/2 + d*x/2...
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {320 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {640 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {45 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 64}{a + \frac {5 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{60 \, d} \]
-1/60*((45*sin(d*x + c)/(cos(d*x + c) + 1) - 320*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 210*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 640*sin(d*x + c)^4/ (cos(d*x + c) + 1)^4 - 210*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 45*sin(d* x + c)^9/(cos(d*x + c) + 1)^9 - 64)/(a + 5*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a*sin(d*x + c)^6/(c os(d*x + c) + 1)^6 + 5*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10) - 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1)) /a)/d
Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {45 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 640 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 64\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} a}}{120 \, d} \]
1/120*(45*(d*x + c)/a + 2*(45*tan(1/2*d*x + 1/2*c)^9 + 210*tan(1/2*d*x + 1 /2*c)^7 + 640*tan(1/2*d*x + 1/2*c)^4 - 210*tan(1/2*d*x + 1/2*c)^3 + 320*ta n(1/2*d*x + 1/2*c)^2 - 45*tan(1/2*d*x + 1/2*c) + 64)/((tan(1/2*d*x + 1/2*c )^2 + 1)^5*a))/d
Time = 13.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^2(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,x}{8\,a}+\frac {\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {16}{15}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]