Integrand size = 21, antiderivative size = 141 \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=-\frac {9 a^2 x}{2}+\frac {2 a^2 \cos (c+d x)}{d}+\frac {6 a^2 \sec (c+d x)}{d}-\frac {2 a^2 \sec ^3(c+d x)}{d}+\frac {2 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {4 a^2 \tan (c+d x)}{d}-\frac {a^2 \tan ^3(c+d x)}{d}+\frac {2 a^2 \tan ^5(c+d x)}{5 d} \] Output:
-9/2*a^2*x+2*a^2*cos(d*x+c)/d+6*a^2*sec(d*x+c)/d-2*a^2*sec(d*x+c)^3/d+2/5* a^2*sec(d*x+c)^5/d+1/2*a^2*cos(d*x+c)*sin(d*x+c)/d+4*a^2*tan(d*x+c)/d-a^2* tan(d*x+c)^3/d+2/5*a^2*tan(d*x+c)^5/d
Time = 1.91 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.23 \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=-\frac {a^2 \sec ^5(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (-500+10 (103+90 c+90 d x) \cos (c+d x)-544 \cos (2 (c+d x))-206 \cos (3 (c+d x))-180 c \cos (3 (c+d x))-180 d x \cos (3 (c+d x))+20 \cos (4 (c+d x))+250 \sin (c+d x)-824 \sin (2 (c+d x))-720 c \sin (2 (c+d x))-720 d x \sin (2 (c+d x))+351 \sin (3 (c+d x))+5 \sin (5 (c+d x)))}{160 d} \] Input:
Integrate[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^6,x]
Output:
-1/160*(a^2*Sec[c + d*x]^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(-500 + 10*(103 + 90*c + 90*d*x)*Cos[c + d*x] - 544*Cos[2*(c + d*x)] - 206*Cos[3* (c + d*x)] - 180*c*Cos[3*(c + d*x)] - 180*d*x*Cos[3*(c + d*x)] + 20*Cos[4* (c + d*x)] + 250*Sin[c + d*x] - 824*Sin[2*(c + d*x)] - 720*c*Sin[2*(c + d* x)] - 720*d*x*Sin[2*(c + d*x)] + 351*Sin[3*(c + d*x)] + 5*Sin[5*(c + d*x)] ))/d
Time = 0.40 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3189, 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.
| \(\displaystyle \int \tan ^6(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^6 (a \sin (c+d x)+a)^2dx\) |
| \(\Big \downarrow \) 3189 |
| \(\displaystyle \int \left (a^2 \tan ^6(c+d x)+a^2 \sin ^2(c+d x) \tan ^6(c+d x)+2 a^2 \sin (c+d x) \tan ^6(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 \cos (c+d x)}{d}+\frac {9 a^2 \tan ^5(c+d x)}{10 d}-\frac {3 a^2 \tan ^3(c+d x)}{2 d}+\frac {9 a^2 \tan (c+d x)}{2 d}+\frac {2 a^2 \sec ^5(c+d x)}{5 d}-\frac {2 a^2 \sec ^3(c+d x)}{d}+\frac {6 a^2 \sec (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x) \tan ^5(c+d x)}{2 d}-\frac {9 a^2 x}{2}\) |
Input:
Int[(a + a*Sin[c + d*x])^2*Tan[c + d*x]^6,x]
Output:
(-9*a^2*x)/2 + (2*a^2*Cos[c + d*x])/d + (6*a^2*Sec[c + d*x])/d - (2*a^2*Se c[c + d*x]^3)/d + (2*a^2*Sec[c + d*x]^5)/(5*d) + (9*a^2*Tan[c + d*x])/(2*d ) - (3*a^2*Tan[c + d*x]^3)/(2*d) + (9*a^2*Tan[c + d*x]^5)/(10*d) - (a^2*Si n[c + d*x]^2*Tan[c + d*x]^5)/(2*d)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*( x_)])^(p_.), x_Symbol] :> Int[ExpandIntegrand[(g*Tan[e + f*x])^p, (a + b*Si n[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Result contains complex when optimal does not.
Time = 5.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(-\frac {9 a^{2} x}{2}-\frac {i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {-\frac {156 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{5}-24 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+\frac {54 i a^{2}}{5}-16 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-30 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{5} d}\) | \(188\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{5 \cos \left (d x +c \right )^{5}}-\frac {4 \sin \left (d x +c \right )^{9}}{15 \cos \left (d x +c \right )^{3}}+\frac {8 \sin \left (d x +c \right )^{9}}{5 \cos \left (d x +c \right )}+\frac {8 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{5}-\frac {7 d x}{2}-\frac {7 c}{2}\right )+2 a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{8}}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+a^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(251\) |
| default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{5 \cos \left (d x +c \right )^{5}}-\frac {4 \sin \left (d x +c \right )^{9}}{15 \cos \left (d x +c \right )^{3}}+\frac {8 \sin \left (d x +c \right )^{9}}{5 \cos \left (d x +c \right )}+\frac {8 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{5}-\frac {7 d x}{2}-\frac {7 c}{2}\right )+2 a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{8}}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+a^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(251\) |
| parts | \(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{5 \cos \left (d x +c \right )^{5}}-\frac {4 \sin \left (d x +c \right )^{9}}{15 \cos \left (d x +c \right )^{3}}+\frac {8 \sin \left (d x +c \right )^{9}}{5 \cos \left (d x +c \right )}+\frac {8 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{5}-\frac {7 d x}{2}-\frac {7 c}{2}\right )}{d}+\frac {2 a^{2} \left (\frac {\sin \left (d x +c \right )^{8}}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{8}}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(258\) |
Input:
int((a+a*sin(d*x+c))^2*tan(d*x+c)^6,x,method=_RETURNVERBOSE)
Output:
-9/2*a^2*x-1/8*I*a^2/d*exp(2*I*(d*x+c))+a^2/d*exp(I*(d*x+c))+a^2/d*exp(-I* (d*x+c))+1/8*I*a^2/d*exp(-2*I*(d*x+c))+2/5*(-78*a^2*exp(I*(d*x+c))-60*I*a^ 2*exp(2*I*(d*x+c))+27*I*a^2-40*a^2*exp(3*I*(d*x+c))-75*I*a^2*exp(4*I*(d*x+ c))+30*a^2*exp(5*I*(d*x+c)))/(exp(I*(d*x+c))+I)/(exp(I*(d*x+c))-I)^5/d
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.08 \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=-\frac {45 \, a^{2} d x \cos \left (d x + c\right )^{3} - 10 \, a^{2} \cos \left (d x + c\right )^{4} - 90 \, a^{2} d x \cos \left (d x + c\right ) + 78 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - {\left (5 \, a^{2} \cos \left (d x + c\right )^{4} - 90 \, a^{2} d x \cos \left (d x + c\right ) + 84 \, a^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{2}\right )} \sin \left (d x + c\right )}{10 \, {\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )\right )}} \] Input:
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^6,x, algorithm="fricas")
Output:
-1/10*(45*a^2*d*x*cos(d*x + c)^3 - 10*a^2*cos(d*x + c)^4 - 90*a^2*d*x*cos( d*x + c) + 78*a^2*cos(d*x + c)^2 - 4*a^2 - (5*a^2*cos(d*x + c)^4 - 90*a^2* d*x*cos(d*x + c) + 84*a^2*cos(d*x + c)^2 - 6*a^2)*sin(d*x + c))/(d*cos(d*x + c)^3 + 2*d*cos(d*x + c)*sin(d*x + c) - 2*d*cos(d*x + c))
\[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=a^{2} \left (\int 2 \sin {\left (c + d x \right )} \tan ^{6}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \tan ^{6}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(d*x+c))**2*tan(d*x+c)**6,x)
Output:
a**2*(Integral(2*sin(c + d*x)*tan(c + d*x)**6, x) + Integral(sin(c + d*x)* *2*tan(c + d*x)**6, x) + Integral(tan(c + d*x)**6, x))
Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.08 \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=\frac {{\left (6 \, \tan \left (d x + c\right )^{5} - 20 \, \tan \left (d x + c\right )^{3} - 105 \, d x - 105 \, c + \frac {15 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} + 90 \, \tan \left (d x + c\right )\right )} a^{2} + 2 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{2} + 12 \, a^{2} {\left (\frac {15 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1}{\cos \left (d x + c\right )^{5}} + 5 \, \cos \left (d x + c\right )\right )}}{30 \, d} \] Input:
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^6,x, algorithm="maxima")
Output:
1/30*((6*tan(d*x + c)^5 - 20*tan(d*x + c)^3 - 105*d*x - 105*c + 15*tan(d*x + c)/(tan(d*x + c)^2 + 1) + 90*tan(d*x + c))*a^2 + 2*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a^2 + 12*a^2*((15*cos( d*x + c)^4 - 5*cos(d*x + c)^2 + 1)/cos(d*x + c)^5 + 5*cos(d*x + c)))/d
Timed out. \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^6,x, algorithm="giac")
Output:
Timed out
Time = 20.31 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.78 \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=-\frac {9\,a^2\,x}{2}-\frac {\frac {9\,a^2\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (180\,c+180\,d\,x-422\right )}{10}\right )+\frac {174\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}-\frac {a^2\,\left (45\,c+45\,d\,x-128\right )}{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (18\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (180\,c+180\,d\,x-90\right )}{10}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (27\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (270\,c+270\,d\,x-168\right )}{10}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {63\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (315\,c+315\,d\,x-360\right )}{10}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (27\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (270\,c+270\,d\,x-600\right )}{10}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (36\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (360\,c+360\,d\,x-424\right )}{10}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {63\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (315\,c+315\,d\,x-536\right )}{10}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (36\,a^2\,\left (c+d\,x\right )-\frac {a^2\,\left (360\,c+360\,d\,x-600\right )}{10}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^5\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \] Input:
int(tan(c + d*x)^6*(a + a*sin(c + d*x))^2,x)
Output:
- (9*a^2*x)/2 - ((9*a^2*(c + d*x))/2 - tan(c/2 + (d*x)/2)*(18*a^2*(c + d*x ) - (a^2*(180*c + 180*d*x - 422))/10) + (174*a^2*tan(c/2 + (d*x)/2)^5)/5 - (a^2*(45*c + 45*d*x - 128))/10 + tan(c/2 + (d*x)/2)^9*(18*a^2*(c + d*x) - (a^2*(180*c + 180*d*x - 90))/10) + tan(c/2 + (d*x)/2)^4*(27*a^2*(c + d*x) - (a^2*(270*c + 270*d*x - 168))/10) - tan(c/2 + (d*x)/2)^8*((63*a^2*(c + d*x))/2 - (a^2*(315*c + 315*d*x - 360))/10) - tan(c/2 + (d*x)/2)^6*(27*a^2 *(c + d*x) - (a^2*(270*c + 270*d*x - 600))/10) - tan(c/2 + (d*x)/2)^3*(36* a^2*(c + d*x) - (a^2*(360*c + 360*d*x - 424))/10) + tan(c/2 + (d*x)/2)^2*( (63*a^2*(c + d*x))/2 - (a^2*(315*c + 315*d*x - 536))/10) + tan(c/2 + (d*x) /2)^7*(36*a^2*(c + d*x) - (a^2*(360*c + 360*d*x - 600))/10))/(d*(tan(c/2 + (d*x)/2) - 1)^5*(tan(c/2 + (d*x)/2) + 1)*(tan(c/2 + (d*x)/2)^2 + 1)^2)
Time = 0.17 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.00 \[ \int (a+a \sin (c+d x))^2 \tan ^6(c+d x) \, dx=\frac {a^{2} \left (192-15 \sin \left (d x +c \right )^{7}-60 \sin \left (d x +c \right )^{6}+161 \sin \left (d x +c \right )^{5}+6 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{5}-105 \cos \left (d x +c \right ) c +384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-480 \sin \left (d x +c \right )^{2}-10 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{3}+30 \cos \left (d x +c \right ) \tan \left (d x +c \right )+360 \sin \left (d x +c \right )^{4}-245 \sin \left (d x +c \right )^{3}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{5}-10 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{3}+30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )-105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} c -12 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{5}+20 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{3}-60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )+210 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} c -135 \cos \left (d x +c \right ) d x -192 \cos \left (d x +c \right )+105 \sin \left (d x +c \right )-135 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} d x +270 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d x -192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}\right )}{30 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:
int((a+a*sin(d*x+c))^2*tan(d*x+c)^6,x)
Output:
(a**2*(6*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x)**5 - 10*cos(c + d*x)*si n(c + d*x)**4*tan(c + d*x)**3 + 30*cos(c + d*x)*sin(c + d*x)**4*tan(c + d* x) - 105*cos(c + d*x)*sin(c + d*x)**4*c - 135*cos(c + d*x)*sin(c + d*x)**4 *d*x - 192*cos(c + d*x)*sin(c + d*x)**4 - 12*cos(c + d*x)*sin(c + d*x)**2* tan(c + d*x)**5 + 20*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**3 - 60*cos (c + d*x)*sin(c + d*x)**2*tan(c + d*x) + 210*cos(c + d*x)*sin(c + d*x)**2* c + 270*cos(c + d*x)*sin(c + d*x)**2*d*x + 384*cos(c + d*x)*sin(c + d*x)** 2 + 6*cos(c + d*x)*tan(c + d*x)**5 - 10*cos(c + d*x)*tan(c + d*x)**3 + 30* cos(c + d*x)*tan(c + d*x) - 105*cos(c + d*x)*c - 135*cos(c + d*x)*d*x - 19 2*cos(c + d*x) - 15*sin(c + d*x)**7 - 60*sin(c + d*x)**6 + 161*sin(c + d*x )**5 + 360*sin(c + d*x)**4 - 245*sin(c + d*x)**3 - 480*sin(c + d*x)**2 + 1 05*sin(c + d*x) + 192))/(30*cos(c + d*x)*d*(sin(c + d*x)**4 - 2*sin(c + d* x)**2 + 1))