Integrand size = 33, antiderivative size = 216 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {(23 A+6 C) x}{2 a^3}+\frac {4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d} \]
-1/2*(23*A+6*C)*x/a^3+4/5*(34*A+9*C)*sin(d*x+c)/a^3/d-1/2*(23*A+6*C)*cos(d *x+c)*sin(d*x+c)/a^3/d-1/5*(A+C)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c) )^3-1/15*(13*A+3*C)*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2-1/3*(23 *A+6*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a^3+a^3*sec(d*x+c))-4/15*(34*A+9*C)*sin (d*x+c)^3/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(216)=432\).
Time = 5.06 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (600 (23 A+6 C) d x \cos \left (\frac {d x}{2}\right )+600 (23 A+6 C) d x \cos \left (c+\frac {d x}{2}\right )+6900 A d x \cos \left (c+\frac {3 d x}{2}\right )+1800 C d x \cos \left (c+\frac {3 d x}{2}\right )+6900 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+1800 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+1380 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+360 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+1380 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+360 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-20410 A \sin \left (\frac {d x}{2}\right )-7020 C \sin \left (\frac {d x}{2}\right )+11110 A \sin \left (c+\frac {d x}{2}\right )+4500 C \sin \left (c+\frac {d x}{2}\right )-15380 A \sin \left (c+\frac {3 d x}{2}\right )-4860 C \sin \left (c+\frac {3 d x}{2}\right )+380 A \sin \left (2 c+\frac {3 d x}{2}\right )+900 C \sin \left (2 c+\frac {3 d x}{2}\right )-4777 A \sin \left (2 c+\frac {5 d x}{2}\right )-1452 C \sin \left (2 c+\frac {5 d x}{2}\right )-1625 A \sin \left (3 c+\frac {5 d x}{2}\right )-300 C \sin \left (3 c+\frac {5 d x}{2}\right )-230 A \sin \left (3 c+\frac {7 d x}{2}\right )-60 C \sin \left (3 c+\frac {7 d x}{2}\right )-230 A \sin \left (4 c+\frac {7 d x}{2}\right )-60 C \sin \left (4 c+\frac {7 d x}{2}\right )+20 A \sin \left (4 c+\frac {9 d x}{2}\right )+20 A \sin \left (5 c+\frac {9 d x}{2}\right )-5 A \sin \left (5 c+\frac {11 d x}{2}\right )-5 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{3840 a^3 d} \]
-1/3840*(Sec[c/2]*Sec[(c + d*x)/2]^5*(600*(23*A + 6*C)*d*x*Cos[(d*x)/2] + 600*(23*A + 6*C)*d*x*Cos[c + (d*x)/2] + 6900*A*d*x*Cos[c + (3*d*x)/2] + 18 00*C*d*x*Cos[c + (3*d*x)/2] + 6900*A*d*x*Cos[2*c + (3*d*x)/2] + 1800*C*d*x *Cos[2*c + (3*d*x)/2] + 1380*A*d*x*Cos[2*c + (5*d*x)/2] + 360*C*d*x*Cos[2* c + (5*d*x)/2] + 1380*A*d*x*Cos[3*c + (5*d*x)/2] + 360*C*d*x*Cos[3*c + (5* d*x)/2] - 20410*A*Sin[(d*x)/2] - 7020*C*Sin[(d*x)/2] + 11110*A*Sin[c + (d* x)/2] + 4500*C*Sin[c + (d*x)/2] - 15380*A*Sin[c + (3*d*x)/2] - 4860*C*Sin[ c + (3*d*x)/2] + 380*A*Sin[2*c + (3*d*x)/2] + 900*C*Sin[2*c + (3*d*x)/2] - 4777*A*Sin[2*c + (5*d*x)/2] - 1452*C*Sin[2*c + (5*d*x)/2] - 1625*A*Sin[3* c + (5*d*x)/2] - 300*C*Sin[3*c + (5*d*x)/2] - 230*A*Sin[3*c + (7*d*x)/2] - 60*C*Sin[3*c + (7*d*x)/2] - 230*A*Sin[4*c + (7*d*x)/2] - 60*C*Sin[4*c + ( 7*d*x)/2] + 20*A*Sin[4*c + (9*d*x)/2] + 20*A*Sin[5*c + (9*d*x)/2] - 5*A*Si n[5*c + (11*d*x)/2] - 5*A*Sin[6*c + (11*d*x)/2]))/(a^3*d)
Time = 1.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 4573, 25, 3042, 4508, 3042, 4508, 27, 3042, 4274, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4573 |
| \(\displaystyle -\frac {\int -\frac {\cos ^3(c+d x) (a (8 A+3 C)-5 a A \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a (8 A+3 C)-5 a A \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (8 A+3 C)-5 a A \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^3(c+d x) \left (9 a^2 (7 A+2 C)-4 a^2 (13 A+3 C) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {9 a^2 (7 A+2 C)-4 a^2 (13 A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int 3 \cos ^3(c+d x) \left (4 a^3 (34 A+9 C)-5 a^3 (23 A+6 C) \sec (c+d x)\right )dx}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \cos ^3(c+d x) \left (4 a^3 (34 A+9 C)-5 a^3 (23 A+6 C) \sec (c+d x)\right )dx}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {4 a^3 (34 A+9 C)-5 a^3 (23 A+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {3 \left (4 a^3 (34 A+9 C) \int \cos ^3(c+d x)dx-5 a^3 (23 A+6 C) \int \cos ^2(c+d x)dx\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (4 a^3 (34 A+9 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx-5 a^3 (23 A+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {4 a^3 (34 A+9 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}-5 a^3 (23 A+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 a^3 (34 A+9 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A+6 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {4 a^3 (34 A+9 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {4 a^3 (34 A+9 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}-5 a^3 (23 A+6 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
-1/5*((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^3) + (- 1/3*(a*(13*A + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^2 ) + ((-5*a^2*(23*A + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d *x])) + (3*(-5*a^3*(23*A + 6*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (4*a^3*(34*A + 9*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d))/a^2)/(3*a^2) )/(5*a^2)
3.2.45.3.1 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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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[(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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-a) *(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*C sc[e + f*x])^n*Simp[b*C*n + A*b*(2*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, C, n}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {-15 \left (\left (-\frac {3416 A}{15}-\frac {312 C}{5}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {43 A}{3}-4 C \right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )-\frac {A \cos \left (5 d x +5 c \right )}{3}+\left (-\frac {10916 A}{15}-\frac {972 C}{5}\right ) \cos \left (d x +c \right )-\frac {7783 A}{15}-\frac {696 C}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-11040 x d \left (A +\frac {6 C}{23}\right )}{960 a^{3} d}\) | \(119\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-\frac {17 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {19 A}{3}-C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {11 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (23 A +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-\frac {17 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {19 A}{3}-C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {11 A}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (23 A +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
| risch | \(-\frac {23 A x}{2 a^{3}}-\frac {3 x C}{a^{3}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {3 i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {27 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{3} d}+\frac {27 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{3} d}-\frac {3 i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {2 i \left (225 A \,{\mathrm e}^{4 i \left (d x +c \right )}+90 C \,{\mathrm e}^{4 i \left (d x +c \right )}+810 A \,{\mathrm e}^{3 i \left (d x +c \right )}+300 C \,{\mathrm e}^{3 i \left (d x +c \right )}+1160 A \,{\mathrm e}^{2 i \left (d x +c \right )}+420 C \,{\mathrm e}^{2 i \left (d x +c \right )}+760 A \,{\mathrm e}^{i \left (d x +c \right )}+270 C \,{\mathrm e}^{i \left (d x +c \right )}+197 A +72 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(293\) |
| norman | \(\frac {\frac {\left (23 A +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (23 A +6 C \right ) x}{2 a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{20 a d}-\frac {\left (11 A +6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{15 a d}-\frac {2 \left (19 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}-\frac {\left (23 A +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {\left (23 A +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2 a}-\frac {\left (93 A +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (127 A +39 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 a d}+\frac {\left (199 A +59 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {\left (207 A +52 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{2}}\) | \(296\) |
1/960*(-15*((-3416/15*A-312/5*C)*cos(2*d*x+2*c)+(-43/3*A-4*C)*cos(3*d*x+3* c)+A*cos(4*d*x+4*c)-1/3*A*cos(5*d*x+5*c)+(-10916/15*A-972/5*C)*cos(d*x+c)- 7783/15*A-696/5*C)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^4-11040*x*d*(A+6/ 23*C))/a^3/d
Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (23 \, A + 6 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (23 \, A + 6 \, C\right )} d x - {\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \, A \cos \left (d x + c\right )^{4} + 5 \, {\left (19 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (869 \, A + 234 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (143 \, A + 38 \, C\right )} \cos \left (d x + c\right ) + 544 \, A + 144 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
-1/30*(15*(23*A + 6*C)*d*x*cos(d*x + c)^3 + 45*(23*A + 6*C)*d*x*cos(d*x + c)^2 + 45*(23*A + 6*C)*d*x*cos(d*x + c) + 15*(23*A + 6*C)*d*x - (10*A*cos( d*x + c)^5 - 15*A*cos(d*x + c)^4 + 5*(19*A + 6*C)*cos(d*x + c)^3 + (869*A + 234*C)*cos(d*x + c)^2 + 9*(143*A + 38*C)*cos(d*x + c) + 544*A + 144*C)*s in(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos( d*x + c) + a^3*d)
\[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
(Integral(A*cos(c + d*x)**3/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(C*cos(c + d*x)**3*sec(c + d*x)**2/(sec(c + d*x )**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x))/a**3
Time = 0.29 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {A {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, C {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
1/60*(A*(20*(33*sin(d*x + c)/(cos(d*x + c) + 1) + 76*sin(d*x + c)^3/(cos(d *x + c) + 1)^3 + 51*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^3 + 3*a^3*sin( d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^ 4 + a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (735*sin(d*x + c)/(cos(d*x + c) + 1) - 50*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos (d*x + c) + 1)^5)/a^3 - 1380*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) + 3*C*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(c os(d*x + c) + 1)) + (85*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^ 3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 120*ar ctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3))/d
Time = 0.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (d x + c\right )} {\left (23 \, A + 6 \, C\right )}}{a^{3}} - \frac {20 \, {\left (51 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 50 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
-1/60*(30*(d*x + c)*(23*A + 6*C)/a^3 - 20*(51*A*tan(1/2*d*x + 1/2*c)^5 + 6 *C*tan(1/2*d*x + 1/2*c)^5 + 76*A*tan(1/2*d*x + 1/2*c)^3 + 12*C*tan(1/2*d*x + 1/2*c)^3 + 33*A*tan(1/2*d*x + 1/2*c) + 6*C*tan(1/2*d*x + 1/2*c))/((tan( 1/2*d*x + 1/2*c)^2 + 1)^3*a^3) - (3*A*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^ 12*tan(1/2*d*x + 1/2*c)^5 - 50*A*a^12*tan(1/2*d*x + 1/2*c)^3 - 30*C*a^12*t an(1/2*d*x + 1/2*c)^3 + 735*A*a^12*tan(1/2*d*x + 1/2*c) + 255*C*a^12*tan(1 /2*d*x + 1/2*c))/a^15)/d
Time = 15.01 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\left (17\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {76\,A}{3}+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {x\,\left (23\,A+6\,C\right )}{2\,a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{3\,a^3}+\frac {6\,A+2\,C}{12\,a^3}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^3}+\frac {6\,A+2\,C}{a^3}+\frac {15\,A-C}{4\,a^3}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \]
(tan(c/2 + (d*x)/2)^5*(17*A + 2*C) + tan(c/2 + (d*x)/2)^3*((76*A)/3 + 4*C) + tan(c/2 + (d*x)/2)*(11*A + 2*C))/(d*(3*a^3*tan(c/2 + (d*x)/2)^2 + 3*a^3 *tan(c/2 + (d*x)/2)^4 + a^3*tan(c/2 + (d*x)/2)^6 + a^3)) - (x*(23*A + 6*C) )/(2*a^3) - (tan(c/2 + (d*x)/2)^3*((A + C)/(3*a^3) + (6*A + 2*C)/(12*a^3)) )/d + (tan(c/2 + (d*x)/2)*((5*(A + C))/(2*a^3) + (6*A + 2*C)/a^3 + (15*A - C)/(4*a^3)))/d + (tan(c/2 + (d*x)/2)^5*(A + C))/(20*a^3*d)