Integrand size = 21, antiderivative size = 98 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {(5 A+6 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {(5 A+6 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {(5 A+6 C) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {A \sec ^5(c+d x) \tan (c+d x)}{6 d} \] Output:
1/16*(5*A+6*C)*arctanh(sin(d*x+c))/d+1/16*(5*A+6*C)*sec(d*x+c)*tan(d*x+c)/ d+1/24*(5*A+6*C)*sec(d*x+c)^3*tan(d*x+c)/d+1/6*A*sec(d*x+c)^5*tan(d*x+c)/d
Time = 0.04 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.40 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {5 A \text {arctanh}(\sin (c+d x))}{16 d}+\frac {3 C \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 A \sec (c+d x) \tan (c+d x)}{16 d}+\frac {3 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 A \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {A \sec ^5(c+d x) \tan (c+d x)}{6 d} \] Input:
Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
Output:
(5*A*ArcTanh[Sin[c + d*x]])/(16*d) + (3*C*ArcTanh[Sin[c + d*x]])/(8*d) + ( 5*A*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (3*C*Sec[c + d*x]*Tan[c + d*x])/(8 *d) + (5*A*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)
Time = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3491, 3042, 4255, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \sec ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3491 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \int \sec ^5(c+d x)dx+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \int \sec ^3(c+d x)dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x)}{6 d}\) |
Input:
Int[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
Output:
(A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((5*A + 6*C)*((Sec[c + d*x]^3*Tan[ c + d*x])/(4*d) + (3*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4))/6
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x _)]^2), x_Symbol] :> Simp[A*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Simp[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)) Int[(b*Sin[e + f* x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.77 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(108\) |
| default | \(\frac {A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(108\) |
| parts | \(\frac {A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(110\) |
| parallelrisch | \(\frac {-225 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {6 C}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+225 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {6 C}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (170 A +204 C \right ) \sin \left (3 d x +3 c \right )+\left (30 A +36 C \right ) \sin \left (5 d x +5 c \right )+396 \left (A +\frac {14 C}{33}\right ) \sin \left (d x +c \right )}{48 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(193\) |
| risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (15 A \,{\mathrm e}^{10 i \left (d x +c \right )}+18 C \,{\mathrm e}^{10 i \left (d x +c \right )}+85 A \,{\mathrm e}^{8 i \left (d x +c \right )}+102 C \,{\mathrm e}^{8 i \left (d x +c \right )}+198 A \,{\mathrm e}^{6 i \left (d x +c \right )}+84 C \,{\mathrm e}^{6 i \left (d x +c \right )}-198 A \,{\mathrm e}^{4 i \left (d x +c \right )}-84 C \,{\mathrm e}^{4 i \left (d x +c \right )}-85 A \,{\mathrm e}^{2 i \left (d x +c \right )}-102 C \,{\mathrm e}^{2 i \left (d x +c \right )}-15 A -18 C \right )}{24 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {5 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}+\frac {5 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}\) | \(255\) |
| norman | \(\frac {\frac {\left (11 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (11 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8 d}+\frac {7 \left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{24 d}+\frac {7 \left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{24 d}+\frac {\left (71 A +18 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {\left (71 A +18 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24 d}+\frac {\left (275 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{24 d}+\frac {\left (275 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {\left (5 A +6 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (5 A +6 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(264\) |
Input:
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x,method=_RETURNVERBOSE)
Output:
1/d*(A*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c)+ 5/16*ln(sec(d*x+c)+tan(d*x+c)))+C*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan (d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c))))
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {3 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, A\right )} \sin \left (d x + c\right )}{96 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="fricas")
Output:
1/96*(3*(5*A + 6*C)*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 3*(5*A + 6*C)*c os(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(3*(5*A + 6*C)*cos(d*x + c)^4 + 2 *(5*A + 6*C)*cos(d*x + c)^2 + 8*A)*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \] Input:
integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**7,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {3 \, {\left (5 \, A + 6 \, C\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A + 6 \, C\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (5 \, A + 6 \, C\right )} \sin \left (d x + c\right )^{5} - 8 \, {\left (5 \, A + 6 \, C\right )} \sin \left (d x + c\right )^{3} + 3 \, {\left (11 \, A + 10 \, C\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1}}{96 \, d} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="maxima")
Output:
1/96*(3*(5*A + 6*C)*log(sin(d*x + c) + 1) - 3*(5*A + 6*C)*log(sin(d*x + c) - 1) - 2*(3*(5*A + 6*C)*sin(d*x + c)^5 - 8*(5*A + 6*C)*sin(d*x + c)^3 + 3 *(11*A + 10*C)*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d* x + c)^2 - 1))/d
Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {3 \, {\left (5 \, A + 6 \, C\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \, {\left (5 \, A + 6 \, C\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, A \sin \left (d x + c\right )^{5} + 18 \, C \sin \left (d x + c\right )^{5} - 40 \, A \sin \left (d x + c\right )^{3} - 48 \, C \sin \left (d x + c\right )^{3} + 33 \, A \sin \left (d x + c\right ) + 30 \, C \sin \left (d x + c\right )\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{96 \, d} \] Input:
integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="giac")
Output:
1/96*(3*(5*A + 6*C)*log(abs(sin(d*x + c) + 1)) - 3*(5*A + 6*C)*log(abs(sin (d*x + c) - 1)) - 2*(15*A*sin(d*x + c)^5 + 18*C*sin(d*x + c)^5 - 40*A*sin( d*x + c)^3 - 48*C*sin(d*x + c)^3 + 33*A*sin(d*x + c) + 30*C*sin(d*x + c))/ (sin(d*x + c)^2 - 1)^3)/d
Time = 41.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (\frac {5\,A}{16}+\frac {3\,C}{8}\right )}{d}-\frac {\left (\frac {5\,A}{16}+\frac {3\,C}{8}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-\frac {5\,A}{6}-C\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {11\,A}{16}+\frac {5\,C}{8}\right )\,\sin \left (c+d\,x\right )}{d\,\left ({\sin \left (c+d\,x\right )}^6-3\,{\sin \left (c+d\,x\right )}^4+3\,{\sin \left (c+d\,x\right )}^2-1\right )} \] Input:
int((A + C*cos(c + d*x)^2)/cos(c + d*x)^7,x)
Output:
(atanh(sin(c + d*x))*((5*A)/16 + (3*C)/8))/d - (sin(c + d*x)*((11*A)/16 + (5*C)/8) - sin(c + d*x)^3*((5*A)/6 + C) + sin(c + d*x)^5*((5*A)/16 + (3*C) /8))/(d*(3*sin(c + d*x)^2 - 3*sin(c + d*x)^4 + sin(c + d*x)^6 - 1))
Time = 0.17 (sec) , antiderivative size = 436, normalized size of antiderivative = 4.45 \[ \int \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {-15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6} a -18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{6} c +45 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} a +54 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} c -45 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a -54 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} c +15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a +18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) c +15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6} a +18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{6} c -45 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} a -54 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} c +45 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a +54 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} c -15 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a -18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) c -15 \sin \left (d x +c \right )^{5} a -18 \sin \left (d x +c \right )^{5} c +40 \sin \left (d x +c \right )^{3} a +48 \sin \left (d x +c \right )^{3} c -33 \sin \left (d x +c \right ) a -30 \sin \left (d x +c \right ) c}{48 d \left (\sin \left (d x +c \right )^{6}-3 \sin \left (d x +c \right )^{4}+3 \sin \left (d x +c \right )^{2}-1\right )} \] Input:
int((A+C*cos(d*x+c)^2)*sec(d*x+c)^7,x)
Output:
( - 15*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a - 18*log(tan((c + d*x)/ 2) - 1)*sin(c + d*x)**6*c + 45*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a + 54*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*c - 45*log(tan((c + d*x)/2 ) - 1)*sin(c + d*x)**2*a - 54*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*c + 15*log(tan((c + d*x)/2) - 1)*a + 18*log(tan((c + d*x)/2) - 1)*c + 15*log (tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a + 18*log(tan((c + d*x)/2) + 1)*si n(c + d*x)**6*c - 45*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a - 54*log( tan((c + d*x)/2) + 1)*sin(c + d*x)**4*c + 45*log(tan((c + d*x)/2) + 1)*sin (c + d*x)**2*a + 54*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*c - 15*log(t an((c + d*x)/2) + 1)*a - 18*log(tan((c + d*x)/2) + 1)*c - 15*sin(c + d*x)* *5*a - 18*sin(c + d*x)**5*c + 40*sin(c + d*x)**3*a + 48*sin(c + d*x)**3*c - 33*sin(c + d*x)*a - 30*sin(c + d*x)*c)/(48*d*(sin(c + d*x)**6 - 3*sin(c + d*x)**4 + 3*sin(c + d*x)**2 - 1))