Integrand size = 33, antiderivative size = 229 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{8} \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) x+\frac {4 a^3 A b \text {arctanh}(\sin (c+d x))}{d}-\frac {a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {a b (12 A-7 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d} \]
1/8*(8*a^4*C+24*a^2*b^2*(2*A+C)+b^4*(4*A+3*C))*x+4*a^3*A*b*arctanh(sin(d*x +c))/d-1/6*a*b*(a^2*(12*A-19*C)-8*b^2*(3*A+2*C))*sin(d*x+c)/d-1/24*b^2*(a^ 2*(24*A-26*C)-3*b^2*(4*A+3*C))*cos(d*x+c)*sin(d*x+c)/d-1/12*a*b*(12*A-7*C) *(a+b*cos(d*x+c))^2*sin(d*x+c)/d-1/4*b*(4*A-C)*(a+b*cos(d*x+c))^3*sin(d*x+ c)/d+A*(a+b*cos(d*x+c))^4*tan(d*x+c)/d
Time = 4.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.20 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {12 \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) (c+d x)-384 a^3 A b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+384 a^3 A b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {96 a^4 A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {96 a^4 A \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+96 a b \left (4 A b^2+4 a^2 C+3 b^2 C\right ) \sin (c+d x)+24 b^2 \left (A b^2+\left (6 a^2+b^2\right ) C\right ) \sin (2 (c+d x))+32 a b^3 C \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 d} \]
(12*(8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*(c + d*x) - 384*a^3 *A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 384*a^3*A*b*Log[Cos[(c + d *x)/2] + Sin[(c + d*x)/2]] + (96*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (96*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin [(c + d*x)/2]) + 96*a*b*(4*A*b^2 + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 24*b^ 2*(A*b^2 + (6*a^2 + b^2)*C)*Sin[2*(c + d*x)] + 32*a*b^3*C*Sin[3*(c + d*x)] + 3*b^4*C*Sin[4*(c + d*x)])/(96*d)
Time = 1.73 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3527, 3042, 3528, 3042, 3528, 3042, 3512, 3042, 3502, 27, 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.
| \(\displaystyle \int \sec ^2(c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \int (a+b \cos (c+d x))^3 \left (-b (4 A-C) \cos ^2(c+d x)+a C \cos (c+d x)+4 A b\right ) \sec (c+d x)dx+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (4 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a C \sin \left (c+d x+\frac {\pi }{2}\right )+4 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (c+d x))^2 \left (-a b (12 A-7 C) \cos ^2(c+d x)+\left (4 C a^2+4 A b^2+3 b^2 C\right ) \cos (c+d x)+16 a A b\right ) \sec (c+d x)dx-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-a b (12 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (4 C a^2+4 A b^2+3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+16 a A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (48 A b a^2+\left (12 C a^2+36 A b^2+23 b^2 C\right ) \cos (c+d x) a-b \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (48 A b a^2+\left (12 C a^2+36 A b^2+23 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-b \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (96 A b a^3-4 b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \cos ^2(c+d x) a+3 \left (8 C a^4+24 b^2 (2 A+C) a^2+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {96 A b a^3-4 b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (8 C a^4+24 b^2 (2 A+C) a^2+b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (32 A b a^3+\left (8 C a^4+24 b^2 (2 A+C) a^2+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{d}\right )-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (32 A b a^3+\left (8 C a^4+24 b^2 (2 A+C) a^2+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{d}\right )-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {32 A b a^3+\left (8 C a^4+24 b^2 (2 A+C) a^2+b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{d}\right )-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (32 a^3 A b \int \sec (c+d x)dx+x \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right )\right )-\frac {4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{d}\right )-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (32 a^3 A b \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+x \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right )\right )-\frac {4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{d}\right )-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {32 a^3 A b \text {arctanh}(\sin (c+d x))}{d}+x \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right )\right )-\frac {4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{d}\right )-\frac {b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\) |
-1/4*(b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/d + (-1/3*(a*b*(12* A - 7*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/d + (-1/2*(b^2*(a^2*(24*A - 26*C) - 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/d + (3*((8*a^4*C + 2 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x + (32*a^3*A*b*ArcTanh[Sin[c + d*x ]])/d) - (4*a*b*(a^2*(12*A - 19*C) - 8*b^2*(3*A + 2*C))*Sin[c + d*x])/d)/2 )/3)/4 + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d
3.6.52.3.1 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_.)*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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 6.61 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.86
| method | result | size |
| parts | \(\frac {a^{4} A \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 A a \,b^{3}+4 C \,a^{3} b \right ) \sin \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+C \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {C \,b^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 C a \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(197\) |
| derivativedivides | \(\frac {a^{4} A \tan \left (d x +c \right )+C \,a^{4} \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \sin \left (d x +c \right ) a^{3} b +6 A \,a^{2} b^{2} \left (d x +c \right )+6 C \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \sin \left (d x +c \right ) a \,b^{3}+\frac {4 C a \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,b^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(207\) |
| default | \(\frac {a^{4} A \tan \left (d x +c \right )+C \,a^{4} \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \sin \left (d x +c \right ) a^{3} b +6 A \,a^{2} b^{2} \left (d x +c \right )+6 C \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \sin \left (d x +c \right ) a \,b^{3}+\frac {4 C a \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,b^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(207\) |
| parallelrisch | \(\frac {-768 A \,a^{3} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+768 A \,a^{3} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+384 b \left (\left (A +\frac {5 C}{6}\right ) b^{2}+a^{2} C \right ) a \sin \left (2 d x +2 c \right )+24 b^{2} \left (\left (A +\frac {9 C}{8}\right ) b^{2}+6 a^{2} C \right ) \sin \left (3 d x +3 c \right )+32 C \sin \left (4 d x +4 c \right ) a \,b^{3}+3 C \sin \left (5 d x +5 c \right ) b^{4}+1152 x \left (\frac {\left (\frac {A}{3}+\frac {C}{4}\right ) b^{4}}{4}+a^{2} \left (A +\frac {C}{2}\right ) b^{2}+\frac {C \,a^{4}}{6}\right ) d \cos \left (d x +c \right )+192 \left (\frac {\left (A +C \right ) b^{4}}{8}+\frac {3 C \,a^{2} b^{2}}{4}+a^{4} A \right ) \sin \left (d x +c \right )}{192 d \cos \left (d x +c \right )}\) | \(226\) |
| risch | \(6 x A \,a^{2} b^{2}+\frac {x A \,b^{4}}{2}+a^{4} C x +3 x C \,a^{2} b^{2}+\frac {3 b^{4} C x}{8}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C \,b^{4}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C \,b^{4}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} A a \,b^{3}}{d}+\frac {2 i a^{4} A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{3} b}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C a \,b^{3}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C a \,b^{3}}{2 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} A a \,b^{3}}{d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A \,b^{4}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A \,b^{4}}{8 d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} C \,a^{2} b^{2}}{4 d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} C \,a^{2} b^{2}}{4 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{3} b}{d}+\frac {4 A \,a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {4 A \,a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (4 d x +4 c \right ) C \,b^{4}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) C a \,b^{3}}{3 d}\) | \(388\) |
| norman | \(\frac {\left (-6 A \,a^{2} b^{2}-\frac {1}{2} A \,b^{4}-C \,a^{4}-3 C \,a^{2} b^{2}-\frac {3}{8} C \,b^{4}\right ) x +\left (-54 A \,a^{2} b^{2}-\frac {9}{2} A \,b^{4}-9 C \,a^{4}-27 C \,a^{2} b^{2}-\frac {27}{8} C \,b^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-30 A \,a^{2} b^{2}-\frac {5}{2} A \,b^{4}-5 C \,a^{4}-15 C \,a^{2} b^{2}-\frac {15}{8} C \,b^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-30 A \,a^{2} b^{2}-\frac {5}{2} A \,b^{4}-5 C \,a^{4}-15 C \,a^{2} b^{2}-\frac {15}{8} C \,b^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{2} b^{2}+\frac {1}{2} A \,b^{4}+C \,a^{4}+3 C \,a^{2} b^{2}+\frac {3}{8} C \,b^{4}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{2} b^{2}+\frac {5}{2} A \,b^{4}+5 C \,a^{4}+15 C \,a^{2} b^{2}+\frac {15}{8} C \,b^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{2} b^{2}+\frac {5}{2} A \,b^{4}+5 C \,a^{4}+15 C \,a^{2} b^{2}+\frac {15}{8} C \,b^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (54 A \,a^{2} b^{2}+\frac {9}{2} A \,b^{4}+9 C \,a^{4}+27 C \,a^{2} b^{2}+\frac {27}{8} C \,b^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (40 a^{4} A -4 A \,b^{4}-24 C \,a^{2} b^{2}-C \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (8 a^{4} A -32 A a \,b^{3}+4 A \,b^{4}-32 C \,a^{3} b +24 C \,a^{2} b^{2}-32 C a \,b^{3}+5 C \,b^{4}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (8 a^{4} A +32 A a \,b^{3}+4 A \,b^{4}+32 C \,a^{3} b +24 C \,a^{2} b^{2}+32 C a \,b^{3}+5 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (72 a^{4} A -192 A a \,b^{3}+12 A \,b^{4}-192 C \,a^{3} b +72 C \,a^{2} b^{2}-128 C a \,b^{3}+3 C \,b^{4}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {\left (72 a^{4} A +192 A a \,b^{3}+12 A \,b^{4}+192 C \,a^{3} b +72 C \,a^{2} b^{2}+128 C a \,b^{3}+3 C \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {\left (360 a^{4} A -480 A a \,b^{3}-12 A \,b^{4}-480 C \,a^{3} b -72 C \,a^{2} b^{2}-224 C a \,b^{3}-15 C \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (360 a^{4} A +480 A a \,b^{3}-12 A \,b^{4}+480 C \,a^{3} b -72 C \,a^{2} b^{2}+224 C a \,b^{3}-15 C \,b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 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 )}-\frac {4 A \,a^{3} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {4 A \,a^{3} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(898\) |
a^4*A/d*tan(d*x+c)+(A*b^4+6*C*a^2*b^2)/d*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d* x+1/2*c)+(4*A*a*b^3+4*C*a^3*b)/d*sin(d*x+c)+(6*A*a^2*b^2+C*a^4)/d*(d*x+c)+ C*b^4/d*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4*A*a ^3*b/d*ln(sec(d*x+c)+tan(d*x+c))+4/3*C*a*b^3/d*(2+cos(d*x+c)^2)*sin(d*x+c)
Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.89 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {48 \, A a^{3} b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, A a^{3} b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (8 \, C a^{4} + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} d x \cos \left (d x + c\right ) + {\left (6 \, C b^{4} \cos \left (d x + c\right )^{4} + 32 \, C a b^{3} \cos \left (d x + c\right )^{3} + 24 \, A a^{4} + 3 \, {\left (24 \, C a^{2} b^{2} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 32 \, {\left (3 \, C a^{3} b + {\left (3 \, A + 2 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
1/24*(48*A*a^3*b*cos(d*x + c)*log(sin(d*x + c) + 1) - 48*A*a^3*b*cos(d*x + c)*log(-sin(d*x + c) + 1) + 3*(8*C*a^4 + 24*(2*A + C)*a^2*b^2 + (4*A + 3* C)*b^4)*d*x*cos(d*x + c) + (6*C*b^4*cos(d*x + c)^4 + 32*C*a*b^3*cos(d*x + c)^3 + 24*A*a^4 + 3*(24*C*a^2*b^2 + (4*A + 3*C)*b^4)*cos(d*x + c)^2 + 32*( 3*C*a^3*b + (3*A + 2*C)*a*b^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) )
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.89 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} C a^{4} + 576 \, {\left (d x + c\right )} A a^{2} b^{2} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b^{2} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{3} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} + 192 \, A a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, C a^{3} b \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \]
1/96*(96*(d*x + c)*C*a^4 + 576*(d*x + c)*A*a^2*b^2 + 144*(2*d*x + 2*c + si n(2*d*x + 2*c))*C*a^2*b^2 - 128*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a*b^3 + 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b^4 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*b^4 + 192*A*a^3*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 384*C*a^3*b*sin(d*x + c) + 384*A*a*b^3*sin(d*x + c) + 96*A*a^4*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (220) = 440\).
Time = 0.38 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.44 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {96 \, A a^{3} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, A a^{3} b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (8 \, C a^{4} + 48 \, A a^{2} b^{2} + 24 \, C a^{2} b^{2} + 4 \, A b^{4} + 3 \, C b^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C b^{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 )}^{4}}}{24 \, d} \]
1/24*(96*A*a^3*b*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 96*A*a^3*b*log(abs(t an(1/2*d*x + 1/2*c) - 1)) - 48*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1 /2*c)^2 - 1) + 3*(8*C*a^4 + 48*A*a^2*b^2 + 24*C*a^2*b^2 + 4*A*b^4 + 3*C*b^ 4)*(d*x + c) + 2*(96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^2*b^2*tan(1/2 *d*x + 1/2*c)^7 + 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 96*C*a*b^3*tan(1/2*d *x + 1/2*c)^7 - 12*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^4*tan(1/2*d*x + 1 /2*c)^7 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 160*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 12*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 9*C*b^4*tan(1/2*d*x + 1/2*c)^ 5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c) ^3 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 160*C*a*b^3*tan(1/2*d*x + 1/2*c) ^3 + 12*A*b^4*tan(1/2*d*x + 1/2*c)^3 - 9*C*b^4*tan(1/2*d*x + 1/2*c)^3 + 96 *C*a^3*b*tan(1/2*d*x + 1/2*c) + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 96*A*a *b^3*tan(1/2*d*x + 1/2*c) + 96*C*a*b^3*tan(1/2*d*x + 1/2*c) + 12*A*b^4*tan (1/2*d*x + 1/2*c) + 15*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 2.38 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.72 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {C\,b^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {4\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {A\,b^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {8\,C\,a\,b^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {4\,C\,a^3\,b\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,b^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {3\,C\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d}+\frac {4\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}-\frac {A\,b^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}}{d}-\frac {C\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,b^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{4\,d}-\frac {A\,a^3\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,8{}\mathrm {i}}{d}-\frac {A\,a^2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,12{}\mathrm {i}}{d}-\frac {C\,a^2\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d} \]
(A*a^4*sin(c + d*x))/(d*cos(c + d*x)) - (C*a^4*atanh((sin(c/2 + (d*x)/2)*1 i)/cos(c/2 + (d*x)/2))*2i)/d - (C*b^4*atanh((sin(c/2 + (d*x)/2)*1i)/cos(c/ 2 + (d*x)/2))*3i)/(4*d) - (A*a^3*b*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*8i)/d - (A*b^4*atanh((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2)) *1i)/d + (C*b^4*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (A*a^2*b^2*atanh((sin (c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*12i)/d - (C*a^2*b^2*atanh((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*6i)/d + (4*A*a*b^3*sin(c + d*x))/d + ( A*b^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (8*C*a*b^3*sin(c + d*x))/(3*d) + (4*C*a^3*b*sin(c + d*x))/d + (3*C*b^4*cos(c + d*x)*sin(c + d*x))/(8*d) + ( 3*C*a^2*b^2*cos(c + d*x)*sin(c + d*x))/d + (4*C*a*b^3*cos(c + d*x)^2*sin(c + d*x))/(3*d)