Integrand size = 33, antiderivative size = 216 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {(6 A+23 C) x}{2 a^3}+\frac {4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d} \] Output:
-1/2*(6*A+23*C)*x/a^3+4/5*(9*A+34*C)*sin(d*x+c)/a^3/d-1/2*(6*A+23*C)*cos(d *x+c)*sin(d*x+c)/a^3/d-1/5*(A+C)*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c) )^3-1/15*(3*A+13*C)*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^2-1/3*(6* A+23*C)*cos(d*x+c)^3*sin(d*x+c)/d/(a^3+a^3*cos(d*x+c))-4/15*(9*A+34*C)*sin (d*x+c)^3/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(463\) vs. \(2(216)=432\).
Time = 3.70 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.14 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (600 (6 A+23 C) d x \cos \left (\frac {d x}{2}\right )+600 (6 A+23 C) d x \cos \left (c+\frac {d x}{2}\right )+1800 A d x \cos \left (c+\frac {3 d x}{2}\right )+6900 C d x \cos \left (c+\frac {3 d x}{2}\right )+1800 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+6900 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+1380 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+1380 C d x \cos \left (3 c+\frac {5 d x}{2}\right )-7020 A \sin \left (\frac {d x}{2}\right )-20410 C \sin \left (\frac {d x}{2}\right )+4500 A \sin \left (c+\frac {d x}{2}\right )+11110 C \sin \left (c+\frac {d x}{2}\right )-4860 A \sin \left (c+\frac {3 d x}{2}\right )-15380 C \sin \left (c+\frac {3 d x}{2}\right )+900 A \sin \left (2 c+\frac {3 d x}{2}\right )+380 C \sin \left (2 c+\frac {3 d x}{2}\right )-1452 A \sin \left (2 c+\frac {5 d x}{2}\right )-4777 C \sin \left (2 c+\frac {5 d x}{2}\right )-300 A \sin \left (3 c+\frac {5 d x}{2}\right )-1625 C \sin \left (3 c+\frac {5 d x}{2}\right )-60 A \sin \left (3 c+\frac {7 d x}{2}\right )-230 C \sin \left (3 c+\frac {7 d x}{2}\right )-60 A \sin \left (4 c+\frac {7 d x}{2}\right )-230 C \sin \left (4 c+\frac {7 d x}{2}\right )+20 C \sin \left (4 c+\frac {9 d x}{2}\right )+20 C \sin \left (5 c+\frac {9 d x}{2}\right )-5 C \sin \left (5 c+\frac {11 d x}{2}\right )-5 C \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{480 a^3 d (1+\cos (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x ]
Output:
-1/480*(Cos[(c + d*x)/2]*Sec[c/2]*(600*(6*A + 23*C)*d*x*Cos[(d*x)/2] + 600 *(6*A + 23*C)*d*x*Cos[c + (d*x)/2] + 1800*A*d*x*Cos[c + (3*d*x)/2] + 6900* C*d*x*Cos[c + (3*d*x)/2] + 1800*A*d*x*Cos[2*c + (3*d*x)/2] + 6900*C*d*x*Co s[2*c + (3*d*x)/2] + 360*A*d*x*Cos[2*c + (5*d*x)/2] + 1380*C*d*x*Cos[2*c + (5*d*x)/2] + 360*A*d*x*Cos[3*c + (5*d*x)/2] + 1380*C*d*x*Cos[3*c + (5*d*x )/2] - 7020*A*Sin[(d*x)/2] - 20410*C*Sin[(d*x)/2] + 4500*A*Sin[c + (d*x)/2 ] + 11110*C*Sin[c + (d*x)/2] - 4860*A*Sin[c + (3*d*x)/2] - 15380*C*Sin[c + (3*d*x)/2] + 900*A*Sin[2*c + (3*d*x)/2] + 380*C*Sin[2*c + (3*d*x)/2] - 14 52*A*Sin[2*c + (5*d*x)/2] - 4777*C*Sin[2*c + (5*d*x)/2] - 300*A*Sin[3*c + (5*d*x)/2] - 1625*C*Sin[3*c + (5*d*x)/2] - 60*A*Sin[3*c + (7*d*x)/2] - 230 *C*Sin[3*c + (7*d*x)/2] - 60*A*Sin[4*c + (7*d*x)/2] - 230*C*Sin[4*c + (7*d *x)/2] + 20*C*Sin[4*c + (9*d*x)/2] + 20*C*Sin[5*c + (9*d*x)/2] - 5*C*Sin[5 *c + (11*d*x)/2] - 5*C*Sin[6*c + (11*d*x)/2]))/(a^3*d*(1 + Cos[c + d*x])^3 )
Time = 1.20 (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, 3521, 25, 3042, 3456, 3042, 3456, 27, 3042, 3227, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int -\frac {\cos ^4(c+d x) (5 a C-a (3 A+8 C) \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\cos ^4(c+d x) (5 a C-a (3 A+8 C) \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (5 a C-a (3 A+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\int \frac {\cos ^3(c+d x) \left (4 a^2 (3 A+13 C)-9 a^2 (2 A+7 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (4 a^2 (3 A+13 C)-9 a^2 (2 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\int 3 \cos ^2(c+d x) \left (5 a^3 (6 A+23 C)-4 a^3 (9 A+34 C) \cos (c+d x)\right )dx}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \cos ^2(c+d x) \left (5 a^3 (6 A+23 C)-4 a^3 (9 A+34 C) \cos (c+d x)\right )dx}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (5 a^3 (6 A+23 C)-4 a^3 (9 A+34 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (5 a^3 (6 A+23 C) \int \cos ^2(c+d x)dx-4 a^3 (9 A+34 C) \int \cos ^3(c+d x)dx\right )}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (5 a^3 (6 A+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-4 a^3 (9 A+34 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {4 a^3 (9 A+34 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}+5 a^3 (6 A+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (5 a^3 (6 A+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {4 a^3 (9 A+34 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (5 a^3 (6 A+23 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {4 a^3 (9 A+34 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}+\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {\frac {5 a^2 (6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac {3 \left (\frac {4 a^3 (9 A+34 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}+5 a^3 (6 A+23 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}}{3 a^2}+\frac {a (3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
Input:
Int[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]
Output:
-1/5*((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^3) - (( a*(3*A + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((5*a^2*(6*A + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]) ) + (3*(5*a^3*(6*A + 23*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) + (4* a^3*(9*A + 34*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d))/a^2)/(3*a^2))/(5* a^2)
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{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] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
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[a*(A + C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(m + n + 2) + C*(b* c*(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Time = 0.56 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {\frac {243 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\frac {2 \left (13 A +\frac {427 C}{9}\right ) \cos \left (2 d x +2 c \right )}{81}+\frac {5 \left (A +\frac {43 C}{12}\right ) \cos \left (3 d x +3 c \right )}{243}-\frac {5 C \cos \left (4 d x +4 c \right )}{972}+\frac {5 C \cos \left (5 d x +5 c \right )}{2916}+\left (A +\frac {2729 C}{729}\right ) \cos \left (d x +c \right )+\frac {58 A}{81}+\frac {7783 C}{2916}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{80}-3 x \left (A +\frac {23 C}{6}\right ) d}{a^{3} d}\) | \(118\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+17 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-\frac {A}{2}-\frac {17 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-A -\frac {19 C}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {A}{2}-\frac {11 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (6 A +23 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+17 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-\frac {A}{2}-\frac {17 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-A -\frac {19 C}{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {A}{2}-\frac {11 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (6 A +23 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(182\) |
| risch | \(-\frac {3 x A}{a^{3}}-\frac {23 C x}{2 a^{3}}-\frac {i C \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {3 i C \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a^{3} d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} C}{8 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a^{3} d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} C}{8 a^{3} d}-\frac {3 i C \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i C \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {2 i \left (90 A \,{\mathrm e}^{4 i \left (d x +c \right )}+225 C \,{\mathrm e}^{4 i \left (d x +c \right )}+300 A \,{\mathrm e}^{3 i \left (d x +c \right )}+810 C \,{\mathrm e}^{3 i \left (d x +c \right )}+420 A \,{\mathrm e}^{2 i \left (d x +c \right )}+1160 C \,{\mathrm e}^{2 i \left (d x +c \right )}+270 A \,{\mathrm e}^{i \left (d x +c \right )}+760 C \,{\mathrm e}^{i \left (d x +c \right )}+72 A +197 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(293\) |
| norman | \(\frac {\frac {\left (21 A +79 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{a d}+\frac {\left (35 A +131 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}-\frac {\left (6 A +23 C \right ) x}{2 a}+\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{20 a d}+\frac {2 \left (A +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{a d}-\frac {\left (3 A +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{15 a d}-\frac {3 \left (6 A +23 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {15 \left (6 A +23 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}-\frac {10 \left (6 A +23 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {15 \left (6 A +23 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2 a}-\frac {3 \left (6 A +23 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a}-\frac {\left (6 A +23 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{2 a}+\frac {\left (25 A +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {4 \left (101 A +381 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 a d}+\frac {3 \left (163 A +618 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 a d}+\frac {\left (387 A +1465 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} a^{2}}\) | \(402\) |
Input:
int(cos(d*x+c)^4*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x,method=_RETURNVER BOSE)
Output:
3/80*(81*tan(1/2*d*x+1/2*c)*(2/81*(13*A+427/9*C)*cos(2*d*x+2*c)+5/243*(A+4 3/12*C)*cos(3*d*x+3*c)-5/972*C*cos(4*d*x+4*c)+5/2916*C*cos(5*d*x+5*c)+(A+2 729/729*C)*cos(d*x+c)+58/81*A+7783/2916*C)*sec(1/2*d*x+1/2*c)^4-80*x*(A+23 /6*C)*d)/a^3/d
Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {15 \, {\left (6 \, A + 23 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (6 \, A + 23 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (6 \, A + 23 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (6 \, A + 23 \, C\right )} d x - {\left (10 \, C \cos \left (d x + c\right )^{5} - 15 \, C \cos \left (d x + c\right )^{4} + 5 \, {\left (6 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (234 \, A + 869 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (38 \, A + 143 \, C\right )} \cos \left (d x + c\right ) + 144 \, A + 544 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)^4*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, algorithm= "fricas")
Output:
-1/30*(15*(6*A + 23*C)*d*x*cos(d*x + c)^3 + 45*(6*A + 23*C)*d*x*cos(d*x + c)^2 + 45*(6*A + 23*C)*d*x*cos(d*x + c) + 15*(6*A + 23*C)*d*x - (10*C*cos( d*x + c)^5 - 15*C*cos(d*x + c)^4 + 5*(6*A + 19*C)*cos(d*x + c)^3 + (234*A + 869*C)*cos(d*x + c)^2 + 9*(38*A + 143*C)*cos(d*x + c) + 144*A + 544*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)
Leaf count of result is larger than twice the leaf count of optimal. 1586 vs. \(2 (201) = 402\).
Time = 6.05 (sec) , antiderivative size = 1586, normalized size of antiderivative = 7.34 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)**4*(A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**3,x)
Output:
Piecewise((-180*A*d*x*tan(c/2 + d*x/2)**6/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3 *d) - 540*A*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a **3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 540*A*d*x*tan(c/2 + d*x/2)**2/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d* tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 180*A* d*x/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180* a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 3*A*tan(c/2 + d*x/2)**11/(60*a** 3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan( c/2 + d*x/2)**2 + 60*a**3*d) - 21*A*tan(c/2 + d*x/2)**9/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2 )**2 + 60*a**3*d) + 174*A*tan(c/2 + d*x/2)**7/(60*a**3*d*tan(c/2 + d*x/2)* *6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60* a**3*d) + 798*A*tan(c/2 + d*x/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a **3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 975*A*tan(c/2 + d*x/2)**3/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan( c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 375*A*tan( c/2 + d*x/2)/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)* *4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 690*C*d*x*tan(c/2 + d*x /2)**6/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 ...
Time = 0.14 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {C {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, A {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, algorithm= "maxima")
Output:
1/60*(C*(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*A*(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 = 0.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (d x + c\right )} {\left (6 \, A + 23 \, C\right )}}{a^{3}} - \frac {20 \, {\left (6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 51 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 50 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, algorithm= "giac")
Output:
-1/60*(30*(d*x + c)*(6*A + 23*C)/a^3 - 20*(6*A*tan(1/2*d*x + 1/2*c)^5 + 51 *C*tan(1/2*d*x + 1/2*c)^5 + 12*A*tan(1/2*d*x + 1/2*c)^3 + 76*C*tan(1/2*d*x + 1/2*c)^3 + 6*A*tan(1/2*d*x + 1/2*c) + 33*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 - 30*A*a^12*tan(1/2*d*x + 1/2*c)^3 - 50*C*a^12*t an(1/2*d*x + 1/2*c)^3 + 255*A*a^12*tan(1/2*d*x + 1/2*c) + 735*C*a^12*tan(1 /2*d*x + 1/2*c))/a^15)/d
Time = 0.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\left (2\,A+17\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A+\frac {76\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A+11\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}-\frac {x\,\left (6\,A+23\,C\right )}{2\,a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{3\,a^3}+\frac {2\,A+6\,C}{12\,a^3}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^3}-\frac {A-15\,C}{4\,a^3}+\frac {2\,A+6\,C}{a^3}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \] Input:
int((cos(c + d*x)^4*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^3,x)
Output:
(tan(c/2 + (d*x)/2)^5*(2*A + 17*C) + tan(c/2 + (d*x)/2)^3*(4*A + (76*C)/3) + tan(c/2 + (d*x)/2)*(2*A + 11*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*(6*A + 23*C) )/(2*a^3) - (tan(c/2 + (d*x)/2)^3*((A + C)/(3*a^3) + (2*A + 6*C)/(12*a^3)) )/d + (tan(c/2 + (d*x)/2)*((5*(A + C))/(2*a^3) - (A - 15*C)/(4*a^3) + (2*A + 6*C)/a^3))/d + (tan(c/2 + (d*x)/2)^5*(A + C))/(20*a^3*d)
Time = 0.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {25 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} c +204 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a +724 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} c -180 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a d x -690 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c d x -6 \cos \left (d x +c \right ) a -6 \cos \left (d x +c \right ) c +10 \sin \left (d x +c \right )^{6} c -30 \sin \left (d x +c \right )^{4} a -140 \sin \left (d x +c \right )^{4} c +90 \sin \left (d x +c \right )^{3} a d x +345 \sin \left (d x +c \right )^{3} c d x +168 \sin \left (d x +c \right )^{2} a +668 \sin \left (d x +c \right )^{2} c -180 \sin \left (d x +c \right ) a d x -690 \sin \left (d x +c \right ) c d x +6 a +6 c}{30 \sin \left (d x +c \right ) a^{3} d \left (2 \cos \left (d x +c \right )-\sin \left (d x +c \right )^{2}+2\right )} \] Input:
int(cos(d*x+c)^4*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x)
Output:
(25*cos(c + d*x)*sin(c + d*x)**4*c + 204*cos(c + d*x)*sin(c + d*x)**2*a + 724*cos(c + d*x)*sin(c + d*x)**2*c - 180*cos(c + d*x)*sin(c + d*x)*a*d*x - 690*cos(c + d*x)*sin(c + d*x)*c*d*x - 6*cos(c + d*x)*a - 6*cos(c + d*x)*c + 10*sin(c + d*x)**6*c - 30*sin(c + d*x)**4*a - 140*sin(c + d*x)**4*c + 9 0*sin(c + d*x)**3*a*d*x + 345*sin(c + d*x)**3*c*d*x + 168*sin(c + d*x)**2* a + 668*sin(c + d*x)**2*c - 180*sin(c + d*x)*a*d*x - 690*sin(c + d*x)*c*d* x + 6*a + 6*c)/(30*sin(c + d*x)*a**3*d*(2*cos(c + d*x) - sin(c + d*x)**2 + 2))