Integrand size = 28, antiderivative size = 216 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {5 a^3 x}{16}+\frac {3}{16} a b^2 x-\frac {a^2 b \cos ^6(c+d x)}{2 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {3 a b^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a b^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a b^2 \cos ^5(c+d x) \sin (c+d x)}{2 d}+\frac {b^3 \sin ^4(c+d x)}{4 d}-\frac {b^3 \sin ^6(c+d x)}{6 d} \]
5/16*a^3*x+3/16*a*b^2*x-1/2*a^2*b*cos(d*x+c)^6/d+5/16*a^3*cos(d*x+c)*sin(d *x+c)/d+3/16*a*b^2*cos(d*x+c)*sin(d*x+c)/d+5/24*a^3*cos(d*x+c)^3*sin(d*x+c )/d+1/8*a*b^2*cos(d*x+c)^3*sin(d*x+c)/d+1/6*a^3*cos(d*x+c)^5*sin(d*x+c)/d- 1/2*a*b^2*cos(d*x+c)^5*sin(d*x+c)/d+1/4*b^3*sin(d*x+c)^4/d-1/6*b^3*sin(d*x +c)^6/d
Time = 1.90 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.79 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {a \left (5 a^2+3 b^2\right ) (c+d x)}{16 d}-\frac {3 b \left (5 a^2+b^2\right ) \cos (2 (c+d x))}{64 d}-\frac {3 a^2 b \cos (4 (c+d x))}{32 d}-\frac {b \left (3 a^2-b^2\right ) \cos (6 (c+d x))}{192 d}+\frac {3 a \left (5 a^2+b^2\right ) \sin (2 (c+d x))}{64 d}+\frac {3 a \left (a^2-b^2\right ) \sin (4 (c+d x))}{64 d}+\frac {a \left (a^2-3 b^2\right ) \sin (6 (c+d x))}{192 d} \]
(a*(5*a^2 + 3*b^2)*(c + d*x))/(16*d) - (3*b*(5*a^2 + b^2)*Cos[2*(c + d*x)] )/(64*d) - (3*a^2*b*Cos[4*(c + d*x)])/(32*d) - (b*(3*a^2 - b^2)*Cos[6*(c + d*x)])/(192*d) + (3*a*(5*a^2 + b^2)*Sin[2*(c + d*x)])/(64*d) + (3*a*(a^2 - b^2)*Sin[4*(c + d*x)])/(64*d) + (a*(a^2 - 3*b^2)*Sin[6*(c + d*x)])/(192* d)
Time = 0.44 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3042, 3569, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^3 \cos ^6(c+d x)+3 a^2 b \sin (c+d x) \cos ^5(c+d x)+3 a b^2 \sin ^2(c+d x) \cos ^4(c+d x)+b^3 \sin ^3(c+d x) \cos ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a^3 x}{16}-\frac {a^2 b \cos ^6(c+d x)}{2 d}-\frac {a b^2 \sin (c+d x) \cos ^5(c+d x)}{2 d}+\frac {a b^2 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {3 a b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {3}{16} a b^2 x-\frac {b^3 \sin ^6(c+d x)}{6 d}+\frac {b^3 \sin ^4(c+d x)}{4 d}\) |
(5*a^3*x)/16 + (3*a*b^2*x)/16 - (a^2*b*Cos[c + d*x]^6)/(2*d) + (5*a^3*Cos[ c + d*x]*Sin[c + d*x])/(16*d) + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*b^2*Cos[c + d*x]^3*Sin[ c + d*x])/(8*d) + (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(2*d) + (b^3*Sin[c + d*x]^4)/(4*d) - (b^3*Sin[c + d* x]^6)/(6*d)
3.1.59.3.1 Defintions of rubi rules used
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*(a *cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte gerQ[m] && IGtQ[n, 0]
Time = 1.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(\frac {\left (-45 a^{2} b -9 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-3 a^{2} b +b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (45 a^{3}+9 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (9 a^{3}-9 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (a^{3}-3 a \,b^{2}\right ) \sin \left (6 d x +6 c \right )+60 a^{3} x d +36 a \,b^{2} d x -18 \cos \left (4 d x +4 c \right ) a^{2} b +66 a^{2} b +8 b^{3}}{192 d}\) | \(154\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {a^{2} b \cos \left (d x +c \right )^{6}}{2}+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{6}-\frac {\cos \left (d x +c \right )^{4}}{12}\right )}{d}\) | \(155\) |
| default | \(\frac {a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {a^{2} b \cos \left (d x +c \right )^{6}}{2}+3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{6}-\frac {\cos \left (d x +c \right )^{4}}{12}\right )}{d}\) | \(155\) |
| parts | \(\frac {a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {b^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}+\frac {\sin \left (d x +c \right )^{4}}{4}\right )}{d}+\frac {3 a \,b^{2} \left (-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}-\frac {a^{2} b \cos \left (d x +c \right )^{6}}{2 d}\) | \(155\) |
| risch | \(\frac {5 a^{3} x}{16}+\frac {3 a \,b^{2} x}{16}-\frac {b \cos \left (6 d x +6 c \right ) a^{2}}{64 d}+\frac {b^{3} \cos \left (6 d x +6 c \right )}{192 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {a \sin \left (6 d x +6 c \right ) b^{2}}{64 d}-\frac {3 b \cos \left (4 d x +4 c \right ) a^{2}}{32 d}+\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {3 a \sin \left (4 d x +4 c \right ) b^{2}}{64 d}-\frac {15 b \cos \left (2 d x +2 c \right ) a^{2}}{64 d}-\frac {3 b^{3} \cos \left (2 d x +2 c \right )}{64 d}+\frac {15 a^{3} \sin \left (2 d x +2 c \right )}{64 d}+\frac {3 a \sin \left (2 d x +2 c \right ) b^{2}}{64 d}\) | \(208\) |
| norman | \(\frac {\left (\frac {5}{16} a^{3}+\frac {3}{16} a \,b^{2}\right ) x +\left (\frac {5}{16} a^{3}+\frac {3}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {15}{8} a^{3}+\frac {9}{8} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {15}{8} a^{3}+\frac {9}{8} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {25}{4} a^{3}+\frac {15}{4} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {75}{16} a^{3}+\frac {45}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {75}{16} a^{3}+\frac {45}{16} a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {4 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {4 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {6 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {6 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}+\frac {4 \left (15 a^{2} b -2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {a \left (5 a^{2}-141 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {a \left (5 a^{2}-141 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {3 a \left (5 a^{2}-13 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {3 a \left (5 a^{2}-13 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {a \left (11 a^{2}-3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a \left (11 a^{2}-3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(454\) |
1/192*((-45*a^2*b-9*b^3)*cos(2*d*x+2*c)+(-3*a^2*b+b^3)*cos(6*d*x+6*c)+(45* a^3+9*a*b^2)*sin(2*d*x+2*c)+(9*a^3-9*a*b^2)*sin(4*d*x+4*c)+(a^3-3*a*b^2)*s in(6*d*x+6*c)+60*a^3*x*d+36*a*b^2*d*x-18*cos(4*d*x+4*c)*a^2*b+66*a^2*b+8*b ^3)/d
Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.59 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=-\frac {12 \, b^{3} \cos \left (d x + c\right )^{4} + 8 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} d x - {\left (8 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]
-1/48*(12*b^3*cos(d*x + c)^4 + 8*(3*a^2*b - b^3)*cos(d*x + c)^6 - 3*(5*a^3 + 3*a*b^2)*d*x - (8*(a^3 - 3*a*b^2)*cos(d*x + c)^5 + 2*(5*a^3 + 3*a*b^2)* cos(d*x + c)^3 + 3*(5*a^3 + 3*a*b^2)*cos(d*x + c))*sin(d*x + c))/d
Time = 0.40 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.85 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {5 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{2} b \cos ^{6}{\left (c + d x \right )}}{2 d} + \frac {3 a b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {b^{3} \sin ^{6}{\left (c + d x \right )}}{12 d} + \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{3} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((5*a**3*x*sin(c + d*x)**6/16 + 15*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*a**3*x*cos( c + d*x)**6/16 + 5*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*a**3*sin(c + d*x)*cos(c + d*x)**5/(1 6*d) - a**2*b*cos(c + d*x)**6/(2*d) + 3*a*b**2*x*sin(c + d*x)**6/16 + 9*a* b**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*a*b**2*x*sin(c + d*x)**2*cos (c + d*x)**4/16 + 3*a*b**2*x*cos(c + d*x)**6/16 + 3*a*b**2*sin(c + d*x)**5 *cos(c + d*x)/(16*d) + a*b**2*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) - 3*a* b**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) + b**3*sin(c + d*x)**6/(12*d) + b **3*sin(c + d*x)**4*cos(c + d*x)**2/(4*d), Ne(d, 0)), (x*(a*cos(c) + b*sin (c))**3*cos(c)**3, True))
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.61 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=-\frac {96 \, a^{2} b \cos \left (d x + c\right )^{6} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 3 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} + 16 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} b^{3}}{192 \, d} \]
-1/192*(96*a^2*b*cos(d*x + c)^6 + (4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^3 - 3*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a*b^2 + 16*(2*sin(d*x + c)^6 - 3*sin(d *x + c)^4)*b^3)/d
Time = 0.38 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.73 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=-\frac {3 \, a^{2} b \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{16} \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {3 \, {\left (5 \, a^{2} b + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, {\left (a^{3} - a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {3 \, {\left (5 \, a^{3} + a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
-3/32*a^2*b*cos(4*d*x + 4*c)/d + 1/16*(5*a^3 + 3*a*b^2)*x - 1/192*(3*a^2*b - b^3)*cos(6*d*x + 6*c)/d - 3/64*(5*a^2*b + b^3)*cos(2*d*x + 2*c)/d + 1/1 92*(a^3 - 3*a*b^2)*sin(6*d*x + 6*c)/d + 3/64*(a^3 - a*b^2)*sin(4*d*x + 4*c )/d + 3/64*(5*a^3 + a*b^2)*sin(2*d*x + 2*c)/d
Time = 24.12 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.88 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a\,b^2}{8}-\frac {11\,a^3}{8}\right )+4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {3\,a\,b^2}{8}-\frac {11\,a^3}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {39\,a\,b^2}{4}-\frac {15\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {39\,a\,b^2}{4}-\frac {15\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {47\,a\,b^2}{8}-\frac {5\,a^3}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {47\,a\,b^2}{8}-\frac {5\,a^3}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (20\,a^2\,b-\frac {8\,b^3}{3}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,a^2+3\,b^2\right )}{8\,\left (\frac {5\,a^3}{8}+\frac {3\,a\,b^2}{8}\right )}\right )\,\left (5\,a^2+3\,b^2\right )}{8\,d}-\frac {a\,\left (5\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \]
(4*b^3*tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)*((3*a*b^2)/8 - (11*a^3)/8 ) + 4*b^3*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^11*((3*a*b^2)/8 - (11* a^3)/8) - tan(c/2 + (d*x)/2)^5*((39*a*b^2)/4 - (15*a^3)/4) + tan(c/2 + (d* x)/2)^7*((39*a*b^2)/4 - (15*a^3)/4) + tan(c/2 + (d*x)/2)^3*((47*a*b^2)/8 - (5*a^3)/24) - tan(c/2 + (d*x)/2)^9*((47*a*b^2)/8 - (5*a^3)/24) + tan(c/2 + (d*x)/2)^6*(20*a^2*b - (8*b^3)/3) + 6*a^2*b*tan(c/2 + (d*x)/2)^2 + 6*a^2 *b*tan(c/2 + (d*x)/2)^10)/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/ 2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d* x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a*atan((a*tan(c/2 + (d*x)/2)*(5* a^2 + 3*b^2))/(8*((3*a*b^2)/8 + (5*a^3)/8)))*(5*a^2 + 3*b^2))/(8*d) - (a*( 5*a^2 + 3*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)