Integrand size = 41, antiderivative size = 239 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(21 A-8 B+2 C) x}{2 a^4}-\frac {8 (216 A-83 B+20 C) \sin (c+d x)}{105 a^4 d}+\frac {(21 A-8 B+2 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A-52 B+10 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {4 (216 A-83 B+20 C) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A-B+C) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3} \] Output:
1/2*(21*A-8*B+2*C)*x/a^4-8/105*(216*A-83*B+20*C)*sin(d*x+c)/a^4/d+1/2*(21* A-8*B+2*C)*cos(d*x+c)*sin(d*x+c)/a^4/d-1/105*(129*A-52*B+10*C)*cos(d*x+c)* sin(d*x+c)/a^4/d/(1+sec(d*x+c))^2-4/105*(216*A-83*B+20*C)*cos(d*x+c)*sin(d *x+c)/a^4/d/(1+sec(d*x+c))-1/7*(A-B+C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d* x+c))^4-1/5*(2*A-B)*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^3
Time = 9.17 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {4 \cos \left (\frac {1}{2} (c+d x)\right ) \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \left (15 (A-B+C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )-6 (39 A-32 B+25 C) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+4 (447 A-286 B+160 C) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )-8 (1653 A-764 B+260 C) \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+210 \cos ^7\left (\frac {1}{2} (c+d x)\right ) (2 (21 A-8 B+2 C) d x+4 (-4 A+B) \sin (c+d x)+A \sin (2 (c+d x)))+15 (A-B+C) \cos \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )-6 (39 A-32 B+25 C) \cos ^3\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+4 (447 A-286 B+160 C) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )\right )}{105 a^4 d (1+\cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \] Input:
Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a* Sec[c + d*x])^4,x]
Output:
(4*Cos[(c + d*x)/2]*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(15*(A - B + C )*Sec[c/2]*Sin[(d*x)/2] - 6*(39*A - 32*B + 25*C)*Cos[(c + d*x)/2]^2*Sec[c/ 2]*Sin[(d*x)/2] + 4*(447*A - 286*B + 160*C)*Cos[(c + d*x)/2]^4*Sec[c/2]*Si n[(d*x)/2] - 8*(1653*A - 764*B + 260*C)*Cos[(c + d*x)/2]^6*Sec[c/2]*Sin[(d *x)/2] + 210*Cos[(c + d*x)/2]^7*(2*(21*A - 8*B + 2*C)*d*x + 4*(-4*A + B)*S in[c + d*x] + A*Sin[2*(c + d*x)]) + 15*(A - B + C)*Cos[(c + d*x)/2]*Tan[c/ 2] - 6*(39*A - 32*B + 25*C)*Cos[(c + d*x)/2]^3*Tan[c/2] + 4*(447*A - 286*B + 160*C)*Cos[(c + d*x)/2]^5*Tan[c/2]))/(105*a^4*d*(1 + Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))
Time = 1.67 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 4572, 3042, 4508, 3042, 4508, 3042, 4508, 3042, 4274, 3042, 3115, 24, 3117}
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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 4572 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (a (9 A-2 B+2 C)-a (5 A-5 B-2 C) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (9 A-2 B+2 C)-a (5 A-5 B-2 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (73 A-24 B+10 C)-28 a^2 (2 A-B) \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 (73 A-24 B+10 C)-28 a^2 (2 A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (a^3 (477 A-176 B+50 C)-3 a^3 (129 A-52 B+10 C) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^3 (477 A-176 B+50 C)-3 a^3 (129 A-52 B+10 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \cos ^2(c+d x) \left (105 a^4 (21 A-8 B+2 C)-8 a^4 (216 A-83 B+20 C) \sec (c+d x)\right )dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {105 a^4 (21 A-8 B+2 C)-8 a^4 (216 A-83 B+20 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \int \cos ^2(c+d x)dx-8 a^4 (216 A-83 B+20 C) \int \cos (c+d x)dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-8 a^4 (216 A-83 B+20 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-8 a^4 (216 A-83 B+20 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-8 a^4 (216 A-83 B+20 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 a^4 (216 A-83 B+20 C) \sin (c+d x)}{d}}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
Input:
Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]
Output:
-1/7*((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^4) + ((-7*a*(2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (-1/3*((129*A - 52*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(d*(1 + Sec[c + d *x])^2) + ((-4*a^3*(216*A - 83*B + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((-8*a^4*(216*A - 83*B + 20*C)*Sin[c + d*x])/d + 105* a^4*(21*A - 8*B + 2*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/a^2)/(3* a^2))/(5*a^2))/(7*a^2)
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B , 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-(a*A - b*B + a*C))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m - n)))*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.70 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.58
| method | result | size |
| parallelrisch | \(\frac {-23619 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {37504 A}{7873}-\frac {43856 B}{23619}+\frac {9920 C}{23619}\right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {9472 B}{23619}+\frac {2080 C}{23619}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {280 A}{7873}-\frac {140 B}{7873}\right ) \cos \left (4 d x +4 c \right )-\frac {35 A \cos \left (5 d x +5 c \right )}{7873}+\left (\frac {85762 A}{7873}-\frac {99968 B}{23619}+\frac {23360 C}{23619}\right ) \cos \left (d x +c \right )+\frac {55656 A}{7873}-\frac {64684 B}{23619}+\frac {15040 C}{23619}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+282240 x d \left (A -\frac {8 B}{21}+\frac {2 C}{21}\right )}{26880 a^{4} d}\) | \(138\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C +13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {16 \left (-\frac {9 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+16 \left (-\frac {7 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A -8 B +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(244\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C +13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {16 \left (-\frac {9 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+16 \left (-\frac {7 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A -8 B +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(244\) |
| norman | \(\frac {-\frac {\left (21 A -8 B +2 C \right ) x}{2 a}+\frac {\left (A -B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{56 a d}-\frac {\left (21 A -8 B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}+\frac {\left (21 A -8 B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}+\frac {\left (21 A -8 B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}-\frac {\left (29 A -22 B +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{140 a d}+\frac {\left (167 A -65 B +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (171 A -62 B +17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}+\frac {\left (1161 A -643 B +265 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{840 a d}-\frac {\left (2529 A -1052 B +275 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{210 a d}-\frac {\left (2913 A -1069 B +265 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\) | \(330\) |
| risch | \(\frac {21 A x}{2 a^{4}}-\frac {4 x B}{a^{4}}+\frac {x C}{a^{4}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{4} d}-\frac {2 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{4} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (2100 A \,{\mathrm e}^{6 i \left (d x +c \right )}-1050 B \,{\mathrm e}^{6 i \left (d x +c \right )}+420 C \,{\mathrm e}^{6 i \left (d x +c \right )}+11025 A \,{\mathrm e}^{5 i \left (d x +c \right )}-5250 B \,{\mathrm e}^{5 i \left (d x +c \right )}+1890 C \,{\mathrm e}^{5 i \left (d x +c \right )}+25515 A \,{\mathrm e}^{4 i \left (d x +c \right )}-11900 B \,{\mathrm e}^{4 i \left (d x +c \right )}+4130 C \,{\mathrm e}^{4 i \left (d x +c \right )}+32340 A \,{\mathrm e}^{3 i \left (d x +c \right )}-14840 B \,{\mathrm e}^{3 i \left (d x +c \right )}+4970 C \,{\mathrm e}^{3 i \left (d x +c \right )}+23688 A \,{\mathrm e}^{2 i \left (d x +c \right )}-10794 B \,{\mathrm e}^{2 i \left (d x +c \right )}+3570 C \,{\mathrm e}^{2 i \left (d x +c \right )}+9471 A \,{\mathrm e}^{i \left (d x +c \right )}-4298 B \,{\mathrm e}^{i \left (d x +c \right )}+1400 C \,{\mathrm e}^{i \left (d x +c \right )}+1653 A -764 B +260 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(384\) |
Input:
int(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x,meth od=_RETURNVERBOSE)
Output:
1/26880*(-23619*tan(1/2*d*x+1/2*c)*((37504/7873*A-43856/23619*B+9920/23619 *C)*cos(2*d*x+2*c)+(A-9472/23619*B+2080/23619*C)*cos(3*d*x+3*c)+(280/7873* A-140/7873*B)*cos(4*d*x+4*c)-35/7873*A*cos(5*d*x+5*c)+(85762/7873*A-99968/ 23619*B+23360/23619*C)*cos(d*x+c)+55656/7873*A-64684/23619*B+15040/23619*C )*sec(1/2*d*x+1/2*c)^6+282240*x*d*(A-8/21*B+2/21*C))/a^4/d
Time = 0.09 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} d x + {\left (105 \, A \cos \left (d x + c\right )^{5} - 210 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (1509 \, A - 592 \, B + 130 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (3411 \, A - 1318 \, B + 310 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (11619 \, A - 4472 \, B + 1070 \, C\right )} \cos \left (d x + c\right ) - 3456 \, A + 1328 \, B - 320 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4, x, algorithm="fricas")
Output:
1/210*(105*(21*A - 8*B + 2*C)*d*x*cos(d*x + c)^4 + 420*(21*A - 8*B + 2*C)* d*x*cos(d*x + c)^3 + 630*(21*A - 8*B + 2*C)*d*x*cos(d*x + c)^2 + 420*(21*A - 8*B + 2*C)*d*x*cos(d*x + c) + 105*(21*A - 8*B + 2*C)*d*x + (105*A*cos(d *x + c)^5 - 210*(2*A - B)*cos(d*x + c)^4 - 4*(1509*A - 592*B + 130*C)*cos( d*x + c)^3 - 4*(3411*A - 1318*B + 310*C)*cos(d*x + c)^2 - (11619*A - 4472* B + 1070*C)*cos(d*x + c) - 3456*A + 1328*B - 320*C)*sin(d*x + c))/(a^4*d*c os(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d* cos(d*x + c) + a^4*d)
\[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(cos(d*x+c)**2*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))* *4,x)
Output:
(Integral(A*cos(c + d*x)**2/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x) + Integral(B*cos(c + d*x)**2*sec(c + d*x)/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x) + Integral(C*cos(c + d*x)**2*sec(c + d*x)**2/(sec(c + d*x)** 4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x))/a**4
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (225) = 450\).
Time = 0.14 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.98 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4, x, algorithm="maxima")
Output:
-1/840*(3*A*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(co s(d*x + c) + 1)^3)/(a^4 + 2*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4* sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) - 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 5880*arctan(sin (d*x + c)/(cos(d*x + c) + 1))/a^4) - B*(1680*sin(d*x + c)/((a^4 + a^4*sin( d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*sin(d*x + c)/ (cos(d*x + c) + 1) - 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 6720*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) + 5*C*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d *x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^ 4 - 336*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4))/d
Time = 0.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (d x + c\right )} {\left (21 \, A - 8 \, B + 2 \, C\right )}}{a^{4}} - \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4, x, algorithm="giac")
Output:
1/840*(420*(d*x + c)*(21*A - 8*B + 2*C)/a^4 - 840*(9*A*tan(1/2*d*x + 1/2*c )^3 - 2*B*tan(1/2*d*x + 1/2*c)^3 + 7*A*tan(1/2*d*x + 1/2*c) - 2*B*tan(1/2* d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^4) + (15*A*a^24*tan(1/2*d* x + 1/2*c)^7 - 15*B*a^24*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 189*A*a^24*tan(1/2*d*x + 1/2*c)^5 + 147*B*a^24*tan(1/2*d*x + 1/ 2*c)^5 - 105*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 1365*A*a^24*tan(1/2*d*x + 1/2 *c)^3 - 805*B*a^24*tan(1/2*d*x + 1/2*c)^3 + 385*C*a^24*tan(1/2*d*x + 1/2*c )^3 - 11655*A*a^24*tan(1/2*d*x + 1/2*c) + 5145*B*a^24*tan(1/2*d*x + 1/2*c) - 1575*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 12.10 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,A-4\,B+2\,C}{8\,a^4}-\frac {5\,B-15\,A+C}{24\,a^4}+\frac {A-B+C}{4\,a^4}\right )}{d}-\frac {\left (9\,A-2\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (7\,A-2\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {6\,A-4\,B+2\,C}{40\,a^4}+\frac {3\,\left (A-B+C\right )}{40\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (6\,A-4\,B+2\,C\right )}{4\,a^4}-\frac {3\,\left (5\,B-15\,A+C\right )}{8\,a^4}+\frac {5\,\left (A-B+C\right )}{4\,a^4}+\frac {20\,A-4\,C}{8\,a^4}\right )}{d}+\frac {x\,\left (21\,A-8\,B+2\,C\right )}{2\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d} \] Input:
int((cos(c + d*x)^2*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^4,x)
Output:
(tan(c/2 + (d*x)/2)^3*((6*A - 4*B + 2*C)/(8*a^4) - (5*B - 15*A + C)/(24*a^ 4) + (A - B + C)/(4*a^4)))/d - (tan(c/2 + (d*x)/2)^3*(9*A - 2*B) + tan(c/2 + (d*x)/2)*(7*A - 2*B))/(d*(2*a^4*tan(c/2 + (d*x)/2)^2 + a^4*tan(c/2 + (d *x)/2)^4 + a^4)) - (tan(c/2 + (d*x)/2)^5*((6*A - 4*B + 2*C)/(40*a^4) + (3* (A - B + C))/(40*a^4)))/d - (tan(c/2 + (d*x)/2)*((3*(6*A - 4*B + 2*C))/(4* a^4) - (3*(5*B - 15*A + C))/(8*a^4) + (5*(A - B + C))/(4*a^4) + (20*A - 4* C)/(8*a^4)))/d + (x*(21*A - 8*B + 2*C))/(2*a^4) + (tan(c/2 + (d*x)/2)^7*(A - B + C))/(56*a^4*d)
Time = 0.16 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.65 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {840 c d x +8820 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a d x -3360 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b d x +840 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} c d x +6825 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +8820 a d x -6720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b d x +17640 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a d x -3360 b d x -1575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) c +15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} b +15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} c -17535 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +1002 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a +190 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} c -9114 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a -910 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} c -29505 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a -2765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} c +11165 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b -526 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b +3682 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b +117 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b -159 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a -75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} c +1680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} c d x}{840 a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x)
Output:
(15*tan((c + d*x)/2)**11*a - 15*tan((c + d*x)/2)**11*b + 15*tan((c + d*x)/ 2)**11*c - 159*tan((c + d*x)/2)**9*a + 117*tan((c + d*x)/2)**9*b - 75*tan( (c + d*x)/2)**9*c + 1002*tan((c + d*x)/2)**7*a - 526*tan((c + d*x)/2)**7*b + 190*tan((c + d*x)/2)**7*c - 9114*tan((c + d*x)/2)**5*a + 3682*tan((c + d*x)/2)**5*b - 910*tan((c + d*x)/2)**5*c + 8820*tan((c + d*x)/2)**4*a*d*x - 3360*tan((c + d*x)/2)**4*b*d*x + 840*tan((c + d*x)/2)**4*c*d*x - 29505*t an((c + d*x)/2)**3*a + 11165*tan((c + d*x)/2)**3*b - 2765*tan((c + d*x)/2) **3*c + 17640*tan((c + d*x)/2)**2*a*d*x - 6720*tan((c + d*x)/2)**2*b*d*x + 1680*tan((c + d*x)/2)**2*c*d*x - 17535*tan((c + d*x)/2)*a + 6825*tan((c + d*x)/2)*b - 1575*tan((c + d*x)/2)*c + 8820*a*d*x - 3360*b*d*x + 840*c*d*x )/(840*a**4*d*(tan((c + d*x)/2)**4 + 2*tan((c + d*x)/2)**2 + 1))