Integrand size = 41, antiderivative size = 206 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {1}{2} a^4 (8 A+13 B+12 C) x+\frac {a^4 (13 A+8 B+2 C) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 a^4 (A-B-2 C) \sin (c+d x)}{2 d}-\frac {(15 A+6 B-2 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{6 d}-\frac {(18 A+3 B-8 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {a (2 A+B) (a+a \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d} \]
1/2*a^4*(8*A+13*B+12*C)*x+1/2*a^4*(13*A+8*B+2*C)*arctanh(sin(d*x+c))/d-5/2 *a^4*(A-B-2*C)*sin(d*x+c)/d-1/6*(15*A+6*B-2*C)*(a^2+a^2*cos(d*x+c))^2*sin( d*x+c)/d-1/6*(18*A+3*B-8*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d+a*(2*A+B)*(a +a*cos(d*x+c))^3*tan(d*x+c)/d+1/2*A*(a+a*cos(d*x+c))^4*sec(d*x+c)*tan(d*x+ c)/d
Time = 9.65 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.45 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {a^4 (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (6 (8 A+13 B+12 C) (c+d x)-6 (13 A+8 B+2 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 (13 A+8 B+2 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 (4 A+B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {3 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 (4 A+B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+3 (4 A+16 B+27 C) \sin (c+d x)+3 (B+4 C) \sin (2 (c+d x))+C \sin (3 (c+d x))\right )}{192 d} \]
(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(6*(8*A + 13*B + 12*C)*(c + d *x) - 6*(13*A + 8*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(1 3*A + 8*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*(4*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2] )^2 + (12*(4*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2] ) + 3*(4*A + 16*B + 27*C)*Sin[c + d*x] + 3*(B + 4*C)*Sin[2*(c + d*x)] + C* Sin[3*(c + d*x)]))/(192*d)
Time = 1.71 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 3522, 3042, 3454, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3214, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (2 a (2 A+B)-a (3 A-2 C) \cos (c+d x)) \sec ^2(c+d x)dx}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (2 a (2 A+B)-a (3 A-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^3 \left (a^2 (13 A+8 B+2 C)-a^2 (15 A+6 B-2 C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a^2 (13 A+8 B+2 C)-a^2 (15 A+6 B-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{3} \int (\cos (c+d x) a+a)^2 \left (3 a^3 (13 A+8 B+2 C)-2 a^3 (18 A+3 B-8 C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 a^3 (13 A+8 B+2 C)-2 a^3 (18 A+3 B-8 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int 6 (\cos (c+d x) a+a) \left (a^4 (13 A+8 B+2 C)-5 a^4 (A-B-2 C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int (\cos (c+d x) a+a) \left (a^4 (13 A+8 B+2 C)-5 a^4 (A-B-2 C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^4 (13 A+8 B+2 C)-5 a^4 (A-B-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \left (-5 (A-B-2 C) \cos ^2(c+d x) a^5+(13 A+8 B+2 C) a^5+\left (a^5 (13 A+8 B+2 C)-5 a^5 (A-B-2 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {-5 (A-B-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(13 A+8 B+2 C) a^5+\left (a^5 (13 A+8 B+2 C)-5 a^5 (A-B-2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\int \left ((13 A+8 B+2 C) a^5+(8 A+13 B+12 C) \cos (c+d x) a^5\right ) \sec (c+d x)dx-\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}\right )-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\int \frac {(13 A+8 B+2 C) a^5+(8 A+13 B+12 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}\right )-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (a^5 (13 A+8 B+2 C) \int \sec (c+d x)dx-\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}+a^5 x (8 A+13 B+12 C)\right )-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (a^5 (13 A+8 B+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}+a^5 x (8 A+13 B+12 C)\right )-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {a^5 (13 A+8 B+2 C) \text {arctanh}(\sin (c+d x))}{d}-\frac {5 a^5 (A-B-2 C) \sin (c+d x)}{d}+a^5 x (8 A+13 B+12 C)\right )-\frac {(18 A+3 B-8 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )-\frac {a^3 (15 A+6 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}+\frac {2 a^2 (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}}{2 a}+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}\) |
(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (-1/3*(a^3*(1 5*A + 6*B - 2*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/d + (-(((18*A + 3*B - 8*C)*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/d) + 3*(a^5*(8*A + 13*B + 12 *C)*x + (a^5*(13*A + 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/d - (5*a^5*(A - B - 2*C)*Sin[c + d*x])/d))/3 + (2*a^2*(2*A + B)*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d)/(2*a)
3.4.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 8.01 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98
| method | result | size |
| parallelrisch | \(\frac {4 a^{4} \left (-\frac {13 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {8 B}{13}+\frac {2 C}{13}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {13 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {8 B}{13}+\frac {2 C}{13}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}+d x \left (\frac {13 B}{8}+\frac {3 C}{2}+A \right ) \cos \left (2 d x +2 c \right )+\left (\frac {C}{4}+\frac {5 B}{16}+A \right ) \sin \left (2 d x +2 c \right )+\frac {\left (B +\frac {83 C}{48}+\frac {A}{4}\right ) \sin \left (3 d x +3 c \right )}{2}+\frac {\left (\frac {B}{4}+C \right ) \sin \left (4 d x +4 c \right )}{8}+\frac {\sin \left (5 d x +5 c \right ) C}{96}+\frac {\left (\frac {3 A}{4}+\frac {41 C}{24}+B \right ) \sin \left (d x +c \right )}{2}+d x \left (\frac {13 B}{8}+\frac {3 C}{2}+A \right )\right )}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(201\) |
| parts | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(218\) |
| derivativedivides | \(\frac {a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+4 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{4} \left (d x +c \right )+6 C \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \,a^{4} \left (d x +c \right )+a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )+C \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(269\) |
| default | \(\frac {a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+4 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{4} \left (d x +c \right )+6 C \,a^{4} \sin \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \,a^{4} \left (d x +c \right )+a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )+C \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(269\) |
| risch | \(4 a^{4} x A +\frac {13 a^{4} B x}{2}+6 a^{4} C x +\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C \,a^{4}}{2 d}-\frac {i a^{4} \left (A \,{\mathrm e}^{3 i \left (d x +c \right )}-8 A \,{\mathrm e}^{2 i \left (d x +c \right )}-2 B \,{\mathrm e}^{2 i \left (d x +c \right )}-A \,{\mathrm e}^{i \left (d x +c \right )}-8 A -2 B \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{4}}{8 d}+\frac {i C \,a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{4}}{8 d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{4}}{8 d}-\frac {i C \,a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{4}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C \,a^{4}}{2 d}-\frac {13 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {13 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}\) | \(458\) |
| norman | \(\frac {\left (4 a^{4} A +\frac {13}{2} B \,a^{4}+6 C \,a^{4}\right ) x +\left (4 a^{4} A +\frac {13}{2} B \,a^{4}+6 C \,a^{4}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-40 a^{4} A -65 B \,a^{4}-60 C \,a^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-16 a^{4} A -26 B \,a^{4}-24 C \,a^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-16 a^{4} A -26 B \,a^{4}-24 C \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (16 a^{4} A +26 B \,a^{4}+24 C \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (16 a^{4} A +26 B \,a^{4}+24 C \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (16 a^{4} A +26 B \,a^{4}+24 C \,a^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (16 a^{4} A +26 B \,a^{4}+24 C \,a^{4}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{4} \left (11 A +11 B +18 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{4} \left (A -B -2 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {7 a^{4} \left (21 A +3 B -10 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (45 A +141 B +158 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a^{4} \left (81 A -39 B -106 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (159 A +105 B +130 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (195 A -99 B -230 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{4} \left (285 A +69 B -2 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a^{4} \left (13 A +8 B +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{4} \left (13 A +8 B +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(598\) |
4*a^4*(-13/8*(1+cos(2*d*x+2*c))*(A+8/13*B+2/13*C)*ln(tan(1/2*d*x+1/2*c)-1) +13/8*(1+cos(2*d*x+2*c))*(A+8/13*B+2/13*C)*ln(tan(1/2*d*x+1/2*c)+1)+d*x*(1 3/8*B+3/2*C+A)*cos(2*d*x+2*c)+(1/4*C+5/16*B+A)*sin(2*d*x+2*c)+1/2*(B+83/48 *C+1/4*A)*sin(3*d*x+3*c)+1/8*(1/4*B+C)*sin(4*d*x+4*c)+1/96*sin(5*d*x+5*c)* C+1/2*(3/4*A+41/24*C+B)*sin(d*x+c)+d*x*(13/8*B+3/2*C+A))/d/(1+cos(2*d*x+2* c))
Time = 0.28 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.93 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {6 \, {\left (8 \, A + 13 \, B + 12 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{2} + 3 \, {\left (13 \, A + 8 \, B + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (13 \, A + 8 \, B + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a^{4} \cos \left (d x + c\right )^{4} + 3 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A + 12 \, B + 20 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 6 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 3 \, A a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3, x, algorithm="fricas")
1/12*(6*(8*A + 13*B + 12*C)*a^4*d*x*cos(d*x + c)^2 + 3*(13*A + 8*B + 2*C)* a^4*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3*(13*A + 8*B + 2*C)*a^4*cos(d* x + c)^2*log(-sin(d*x + c) + 1) + 2*(2*C*a^4*cos(d*x + c)^4 + 3*(B + 4*C)* a^4*cos(d*x + c)^3 + 2*(3*A + 12*B + 20*C)*a^4*cos(d*x + c)^2 + 6*(4*A + B )*a^4*cos(d*x + c) + 3*A*a^4)*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.44 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {48 \, {\left (d x + c\right )} A a^{4} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 72 \, {\left (d x + c\right )} B a^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 48 \, {\left (d x + c\right )} C a^{4} - 3 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{4} \sin \left (d x + c\right ) + 48 \, B a^{4} \sin \left (d x + c\right ) + 72 \, C a^{4} \sin \left (d x + c\right ) + 48 \, A a^{4} \tan \left (d x + c\right ) + 12 \, B a^{4} \tan \left (d x + c\right )}{12 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3, x, algorithm="maxima")
1/12*(48*(d*x + c)*A*a^4 + 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 + 72*( d*x + c)*B*a^4 - 4*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 12*(2*d*x + 2 *c + sin(2*d*x + 2*c))*C*a^4 + 48*(d*x + c)*C*a^4 - 3*A*a^4*(2*sin(d*x + c )/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 36*A*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*B*a^4*(log(s in(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 6*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 12*A*a^4*sin(d*x + c) + 48*B*a^4*sin(d*x + c) + 72*C*a^4*sin(d*x + c) + 48*A*a^4*tan(d*x + c) + 12*B*a^4*tan(d*x + c))/d
Time = 0.40 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.68 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {3 \, {\left (8 \, A a^{4} + 13 \, B a^{4} + 12 \, C a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (13 \, A a^{4} + 8 \, B a^{4} + 2 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (13 \, A a^{4} + 8 \, B a^{4} + 2 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (7 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 54 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3, x, algorithm="giac")
1/6*(3*(8*A*a^4 + 13*B*a^4 + 12*C*a^4)*(d*x + c) + 3*(13*A*a^4 + 8*B*a^4 + 2*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(13*A*a^4 + 8*B*a^4 + 2*C *a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 6*(7*A*a^4*tan(1/2*d*x + 1/2*c) ^3 + 2*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 9*A*a^4*tan(1/2*d*x + 1/2*c) - 2*B*a ^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + 2*(6*A*a^4*tan(1 /2*d*x + 1/2*c)^5 + 21*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 30*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 12*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 48*B*a^4*tan(1/2*d*x + 1/2 *c)^3 + 76*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^4*tan(1/2*d*x + 1/2*c) + 2 7*B*a^4*tan(1/2*d*x + 1/2*c) + 54*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 3.06 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.81 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {2\,\left (4\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-\frac {A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,13{}\mathrm {i}}{2}+\frac {13\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,4{}\mathrm {i}+6\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}\right )}{d}+\frac {2\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {5\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{8}+B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{16}+\frac {C\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {83\,C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48}+\frac {C\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{4}+\frac {C\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{48}+\frac {3\,A\,a^4\,\sin \left (c+d\,x\right )}{4}+B\,a^4\,\sin \left (c+d\,x\right )+\frac {41\,C\,a^4\,\sin \left (c+d\,x\right )}{24}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \]
(2*(4*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) - (A*a^4*atan((sin (c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*13i)/2 + (13*B*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 - B*a^4*atan((sin(c/2 + (d*x)/2)*1i)/cos(c /2 + (d*x)/2))*4i + 6*C*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) - C*a^4*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*1i))/d + (2*A*a^4*s in(2*c + 2*d*x) + (A*a^4*sin(3*c + 3*d*x))/4 + (5*B*a^4*sin(2*c + 2*d*x))/ 8 + B*a^4*sin(3*c + 3*d*x) + (B*a^4*sin(4*c + 4*d*x))/16 + (C*a^4*sin(2*c + 2*d*x))/2 + (83*C*a^4*sin(3*c + 3*d*x))/48 + (C*a^4*sin(4*c + 4*d*x))/4 + (C*a^4*sin(5*c + 5*d*x))/48 + (3*A*a^4*sin(c + d*x))/4 + B*a^4*sin(c + d *x) + (41*C*a^4*sin(c + d*x))/24)/(d*(cos(2*c + 2*d*x)/2 + 1/2))