Integrand size = 41, antiderivative size = 164 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {C x}{a^4}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(16 A+12 B-215 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(4 A+3 B-10 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \] Output:
C*x/a^4-1/105*(8*A+6*B-55*C)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2+1/105*(16*A +12*B-215*C)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*(A-B+C)*cos(d*x+c)^3*sin( d*x+c)/d/(a+a*cos(d*x+c))^4+1/35*(4*A+3*B-10*C)*cos(d*x+c)^2*sin(d*x+c)/a/ d/(a+a*cos(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(405\) vs. \(2(164)=328\).
Time = 7.03 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (3675 C d x \cos \left (\frac {d x}{2}\right )+3675 C d x \cos \left (c+\frac {d x}{2}\right )+2205 C d x \cos \left (c+\frac {3 d x}{2}\right )+2205 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+735 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+735 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+105 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+105 C d x \cos \left (4 c+\frac {7 d x}{2}\right )+560 A \sin \left (\frac {d x}{2}\right )+1260 B \sin \left (\frac {d x}{2}\right )-9940 C \sin \left (\frac {d x}{2}\right )-350 A \sin \left (c+\frac {d x}{2}\right )-1260 B \sin \left (c+\frac {d x}{2}\right )+8260 C \sin \left (c+\frac {d x}{2}\right )+336 A \sin \left (c+\frac {3 d x}{2}\right )+882 B \sin \left (c+\frac {3 d x}{2}\right )-7140 C \sin \left (c+\frac {3 d x}{2}\right )-210 A \sin \left (2 c+\frac {3 d x}{2}\right )-630 B \sin \left (2 c+\frac {3 d x}{2}\right )+3780 C \sin \left (2 c+\frac {3 d x}{2}\right )+182 A \sin \left (2 c+\frac {5 d x}{2}\right )+294 B \sin \left (2 c+\frac {5 d x}{2}\right )-2800 C \sin \left (2 c+\frac {5 d x}{2}\right )-210 B \sin \left (3 c+\frac {5 d x}{2}\right )+840 C \sin \left (3 c+\frac {5 d x}{2}\right )+26 A \sin \left (3 c+\frac {7 d x}{2}\right )+72 B \sin \left (3 c+\frac {7 d x}{2}\right )-520 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{13440 a^4 d} \] Input:
Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a* Cos[c + d*x])^4,x]
Output:
(Sec[c/2]*Sec[(c + d*x)/2]^7*(3675*C*d*x*Cos[(d*x)/2] + 3675*C*d*x*Cos[c + (d*x)/2] + 2205*C*d*x*Cos[c + (3*d*x)/2] + 2205*C*d*x*Cos[2*c + (3*d*x)/2 ] + 735*C*d*x*Cos[2*c + (5*d*x)/2] + 735*C*d*x*Cos[3*c + (5*d*x)/2] + 105* C*d*x*Cos[3*c + (7*d*x)/2] + 105*C*d*x*Cos[4*c + (7*d*x)/2] + 560*A*Sin[(d *x)/2] + 1260*B*Sin[(d*x)/2] - 9940*C*Sin[(d*x)/2] - 350*A*Sin[c + (d*x)/2 ] - 1260*B*Sin[c + (d*x)/2] + 8260*C*Sin[c + (d*x)/2] + 336*A*Sin[c + (3*d *x)/2] + 882*B*Sin[c + (3*d*x)/2] - 7140*C*Sin[c + (3*d*x)/2] - 210*A*Sin[ 2*c + (3*d*x)/2] - 630*B*Sin[2*c + (3*d*x)/2] + 3780*C*Sin[2*c + (3*d*x)/2 ] + 182*A*Sin[2*c + (5*d*x)/2] + 294*B*Sin[2*c + (5*d*x)/2] - 2800*C*Sin[2 *c + (5*d*x)/2] - 210*B*Sin[3*c + (5*d*x)/2] + 840*C*Sin[3*c + (5*d*x)/2] + 26*A*Sin[3*c + (7*d*x)/2] + 72*B*Sin[3*c + (7*d*x)/2] - 520*C*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)
Time = 1.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {3042, 3520, 3042, 3456, 3042, 3447, 3042, 3498, 25, 3042, 3214, 3042, 3127}
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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (a (4 A+3 B-3 C)+7 a C \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a (4 A+3 B-3 C)+7 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {\cos (c+d x) \left (2 (4 A+3 B-10 C) a^2+35 C \cos (c+d x) a^2\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 (4 A+3 B-10 C) a^2+35 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\int \frac {35 C \cos ^2(c+d x) a^2+2 (4 A+3 B-10 C) \cos (c+d x) a^2}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {35 C \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+2 (4 A+3 B-10 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 (8 A+6 B-55 C) a^3+105 C \cos (c+d x) a^3}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 (8 A+6 B-55 C) a^3+105 C \cos (c+d x) a^3}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 (8 A+6 B-55 C) a^3+105 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {a^3 (16 A+12 B-215 C) \int \frac {1}{\cos (c+d x) a+a}dx+105 a^2 C x}{3 a^2}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^3 (16 A+12 B-215 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+105 a^2 C x}{3 a^2}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 (16 A+12 B-215 C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+105 a^2 C x}{3 a^2}-\frac {(8 A+6 B-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {a (4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
Input:
Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]
Output:
-1/7*((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + ((a*(4*A + 3*B - 10*C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (-1/3*((8*A + 6*B - 55*C)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^2 ) + (105*a^2*C*x + (a^3*(16*A + 12*B - 215*C)*Sin[c + d*x])/(d*(a + a*Cos[ c + d*x])))/(3*a^2))/(5*a^2))/(7*a^2)
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
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[(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_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a* B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[1 /(a^2*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b *B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
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[(a*A - b*B + 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)) Int[(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)) + B*(b*c*m + a *d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(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, B, 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 = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {15 \left (A -B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+21 \left (-A +3 B -5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+35 \left (-A -3 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+105 \left (A +B -15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 d x C}{840 a^{4} d}\) | \(97\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -15 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(183\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -15 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(183\) |
| risch | \(\frac {C x}{a^{4}}+\frac {2 i \left (105 B \,{\mathrm e}^{6 i \left (d x +c \right )}-420 C \,{\mathrm e}^{6 i \left (d x +c \right )}+105 A \,{\mathrm e}^{5 i \left (d x +c \right )}+315 B \,{\mathrm e}^{5 i \left (d x +c \right )}-1890 C \,{\mathrm e}^{5 i \left (d x +c \right )}+175 A \,{\mathrm e}^{4 i \left (d x +c \right )}+630 B \,{\mathrm e}^{4 i \left (d x +c \right )}-4130 C \,{\mathrm e}^{4 i \left (d x +c \right )}+280 A \,{\mathrm e}^{3 i \left (d x +c \right )}+630 B \,{\mathrm e}^{3 i \left (d x +c \right )}-4970 C \,{\mathrm e}^{3 i \left (d x +c \right )}+168 A \,{\mathrm e}^{2 i \left (d x +c \right )}+441 B \,{\mathrm e}^{2 i \left (d x +c \right )}-3570 C \,{\mathrm e}^{2 i \left (d x +c \right )}+91 A \,{\mathrm e}^{i \left (d x +c \right )}+147 B \,{\mathrm e}^{i \left (d x +c \right )}-1400 C \,{\mathrm e}^{i \left (d x +c \right )}+13 A +36 B -260 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(244\) |
| norman | \(\frac {\frac {C x}{a}+\frac {C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}+\frac {4 C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {6 C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}+\frac {4 C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}+\frac {\left (A -B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{56 a d}+\frac {\left (A +B -15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\left (11 A +9 B -169 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d a}+\frac {\left (13 A +B -15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{280 d a}-\frac {\left (29 A -57 B -55 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{840 d a}+\frac {\left (47 A +9 B -1465 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{280 d a}+\frac {\left (67 A +39 B -1145 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d a}-\frac {\left (101 A -3 B +605 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{840 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} a^{3}}\) | \(318\) |
Input:
int(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x,meth od=_RETURNVERBOSE)
Output:
1/840*(15*(A-B+C)*tan(1/2*d*x+1/2*c)^7+21*(-A+3*B-5*C)*tan(1/2*d*x+1/2*c)^ 5+35*(-A-3*B+11*C)*tan(1/2*d*x+1/2*c)^3+105*(A+B-15*C)*tan(1/2*d*x+1/2*c)+ 840*d*x*C)/a^4/d
Time = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, C d x \cos \left (d x + c\right )^{4} + 420 \, C d x \cos \left (d x + c\right )^{3} + 630 \, C d x \cos \left (d x + c\right )^{2} + 420 \, C d x \cos \left (d x + c\right ) + 105 \, C d x + {\left ({\left (13 \, A + 36 \, B - 260 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (52 \, A + 39 \, B - 620 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (32 \, A + 24 \, B - 535 \, C\right )} \cos \left (d x + c\right ) + 8 \, A + 6 \, B - 160 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4, x, algorithm="fricas")
Output:
1/105*(105*C*d*x*cos(d*x + c)^4 + 420*C*d*x*cos(d*x + c)^3 + 630*C*d*x*cos (d*x + c)^2 + 420*C*d*x*cos(d*x + c) + 105*C*d*x + ((13*A + 36*B - 260*C)* cos(d*x + c)^3 + (52*A + 39*B - 620*C)*cos(d*x + c)^2 + (32*A + 24*B - 535 *C)*cos(d*x + c) + 8*A + 6*B - 160*C)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
Time = 4.12 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.70 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} - \frac {B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {C x}{a^{4}} + \frac {C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {11 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} - \frac {15 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**2*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))* *4,x)
Output:
Piecewise((A*tan(c/2 + d*x/2)**7/(56*a**4*d) - A*tan(c/2 + d*x/2)**5/(40*a **4*d) - A*tan(c/2 + d*x/2)**3/(24*a**4*d) + A*tan(c/2 + d*x/2)/(8*a**4*d) - B*tan(c/2 + d*x/2)**7/(56*a**4*d) + 3*B*tan(c/2 + d*x/2)**5/(40*a**4*d) - B*tan(c/2 + d*x/2)**3/(8*a**4*d) + B*tan(c/2 + d*x/2)/(8*a**4*d) + C*x/ a**4 + C*tan(c/2 + d*x/2)**7/(56*a**4*d) - C*tan(c/2 + d*x/2)**5/(8*a**4*d ) + 11*C*tan(c/2 + d*x/2)**3/(24*a**4*d) - 15*C*tan(c/2 + d*x/2)/(8*a**4*d ), Ne(d, 0)), (x*(A + B*cos(c) + C*cos(c)**2)*cos(c)**2/(a*cos(c) + a)**4, True))
Time = 0.16 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.74 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {5 \, C {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac {A {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} - \frac {3 \, B {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4, x, algorithm="maxima")
Output:
-1/840*(5*C*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos (d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*sin(d*x + c) ^7/(cos(d*x + c) + 1)^7)/a^4 - 336*arctan(sin(d*x + c)/(cos(d*x + c) + 1)) /a^4) - A*(105*sin(d*x + c)/(cos(d*x + c) + 1) - 35*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 - 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 15*sin(d*x + c)^7 /(cos(d*x + c) + 1)^7)/a^4 - 3*B*(35*sin(d*x + c)/(cos(d*x + c) + 1) - 35* sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1) ^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4)/d
Time = 0.17 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.34 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (d x + c\right )} C}{a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4, x, algorithm="giac")
Output:
1/840*(840*(d*x + c)*C/a^4 + (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 - 15*B*a^24 *tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 21*A*a^24*tan (1/2*d*x + 1/2*c)^5 + 63*B*a^24*tan(1/2*d*x + 1/2*c)^5 - 105*C*a^24*tan(1/ 2*d*x + 1/2*c)^5 - 35*A*a^24*tan(1/2*d*x + 1/2*c)^3 - 105*B*a^24*tan(1/2*d *x + 1/2*c)^3 + 385*C*a^24*tan(1/2*d*x + 1/2*c)^3 + 105*A*a^24*tan(1/2*d*x + 1/2*c) + 105*B*a^24*tan(1/2*d*x + 1/2*c) - 1575*C*a^24*tan(1/2*d*x + 1/ 2*c))/a^28)/d
Time = 0.57 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.40 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {C\,x}{a^4}+\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}-\frac {15\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}-\frac {11\,C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}-\frac {3\,B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}+\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}\right )+\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}-\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}+\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int((cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^4,x)
Output:
(C*x)/a^4 + (cos(c/2 + (d*x)/2)^6*((A*sin(c/2 + (d*x)/2))/8 + (B*sin(c/2 + (d*x)/2))/8 - (15*C*sin(c/2 + (d*x)/2))/8) - cos(c/2 + (d*x)/2)^4*((A*sin (c/2 + (d*x)/2)^3)/24 + (B*sin(c/2 + (d*x)/2)^3)/8 - (11*C*sin(c/2 + (d*x) /2)^3)/24) - cos(c/2 + (d*x)/2)^2*((A*sin(c/2 + (d*x)/2)^5)/40 - (3*B*sin( c/2 + (d*x)/2)^5)/40 + (C*sin(c/2 + (d*x)/2)^5)/8) + (A*sin(c/2 + (d*x)/2) ^7)/56 - (B*sin(c/2 + (d*x)/2)^7)/56 + (C*sin(c/2 + (d*x)/2)^7)/56)/(a^4*d *cos(c/2 + (d*x)/2)^7)
Time = 0.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b +15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} c -21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a +63 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b -105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} c -35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a -105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +385 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} c +105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b -1575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) c +840 c d x}{840 a^{4} d} \] Input:
int(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x)
Output:
(15*tan((c + d*x)/2)**7*a - 15*tan((c + d*x)/2)**7*b + 15*tan((c + d*x)/2) **7*c - 21*tan((c + d*x)/2)**5*a + 63*tan((c + d*x)/2)**5*b - 105*tan((c + d*x)/2)**5*c - 35*tan((c + d*x)/2)**3*a - 105*tan((c + d*x)/2)**3*b + 385 *tan((c + d*x)/2)**3*c + 105*tan((c + d*x)/2)*a + 105*tan((c + d*x)/2)*b - 1575*tan((c + d*x)/2)*c + 840*c*d*x)/(840*a**4*d)