Integrand size = 33, antiderivative size = 152 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {C x}{a^4}-\frac {(8 A-55 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(16 A-215 C) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A+C) \cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (2 A-5 C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
C*x/a^4-1/105*(8*A-55*C)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2+1/105*(16*A-215 *C)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*(A+C)*cos(d*x+c)^3*sin(d*x+c)/d/(a +a*cos(d*x+c))^4+2/35*(2*A-5*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. \(315\) vs. \(2(152)=304\).
Time = 4.81 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.07 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (3675 C d x \cos \left (\frac {d x}{2}\right )+3675 C d x \cos \left (c+\frac {d x}{2}\right )+2205 C d x \cos \left (c+\frac {3 d x}{2}\right )+2205 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+735 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+735 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+105 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+105 C d x \cos \left (4 c+\frac {7 d x}{2}\right )+560 A \sin \left (\frac {d x}{2}\right )-9940 C \sin \left (\frac {d x}{2}\right )-350 A \sin \left (c+\frac {d x}{2}\right )+8260 C \sin \left (c+\frac {d x}{2}\right )+336 A \sin \left (c+\frac {3 d x}{2}\right )-7140 C \sin \left (c+\frac {3 d x}{2}\right )-210 A \sin \left (2 c+\frac {3 d x}{2}\right )+3780 C \sin \left (2 c+\frac {3 d x}{2}\right )+182 A \sin \left (2 c+\frac {5 d x}{2}\right )-2800 C \sin \left (2 c+\frac {5 d x}{2}\right )+840 C \sin \left (3 c+\frac {5 d x}{2}\right )+26 A \sin \left (3 c+\frac {7 d x}{2}\right )-520 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{13440 a^4 d} \]
(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] - 9940*C*Sin[(d*x)/2] - 350*A*Sin[c + (d*x)/2] + 8260*C*Sin[c + (d* x)/2] + 336*A*Sin[c + (3*d*x)/2] - 7140*C*Sin[c + (3*d*x)/2] - 210*A*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] - 2800*C*Sin[2*c + (5*d*x)/2] + 840*C*Sin[3*c + (5*d*x)/2] + 26*A*Sin[3*c + (7*d*x)/2] - 520*C*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)
Time = 1.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 3521, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (a (4 A-3 C)+7 a C \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a (4 A-3 C)+7 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int \frac {\cos (c+d x) \left (4 (2 A-5 C) a^2+35 C \cos (c+d x) a^2\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (4 (2 A-5 C) a^2+35 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\int \frac {35 C \cos ^2(c+d x) a^2+4 (2 A-5 C) \cos (c+d x) a^2}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {35 C \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+4 (2 A-5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 (8 A-55 C) a^3+105 C \cos (c+d x) a^3}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(8 A-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 (8 A-55 C) a^3+105 C \cos (c+d x) a^3}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(8 A-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 (8 A-55 C) a^3+105 C \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {(8 A-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {a^3 (16 A-215 C) \int \frac {1}{\cos (c+d x) a+a}dx+105 a^2 C x}{3 a^2}-\frac {(8 A-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^3 (16 A-215 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+105 a^2 C x}{3 a^2}-\frac {(8 A-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {\frac {\frac {\frac {a^3 (16 A-215 C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+105 a^2 C x}{3 a^2}-\frac {(8 A-55 C) \sin (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + (( 2*a*(2*A - 5*C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (-1/3*((8*A - 55*C)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^2) + (105*a^2*C* x + (a^3*(16*A - 215*C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])))/(3*a^2))/( 5*a^2))/(7*a^2)
3.1.67.3.1 Defintions of rubi rules used
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_.) + (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 = 1.66 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.57
| method | result | size |
| parallelrisch | \(\frac {15 \left (A +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \left (-A -5 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (-A +11 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \left (A -15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 d x C}{840 a^{4} d}\) | \(87\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +16 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(130\) |
| default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +16 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(130\) |
| risch | \(\frac {C x}{a^{4}}-\frac {2 i \left (420 C \,{\mathrm e}^{6 i \left (d x +c \right )}-105 A \,{\mathrm e}^{5 i \left (d x +c \right )}+1890 C \,{\mathrm e}^{5 i \left (d x +c \right )}-175 A \,{\mathrm e}^{4 i \left (d x +c \right )}+4130 C \,{\mathrm e}^{4 i \left (d x +c \right )}-280 A \,{\mathrm e}^{3 i \left (d x +c \right )}+4970 C \,{\mathrm e}^{3 i \left (d x +c \right )}-168 A \,{\mathrm e}^{2 i \left (d x +c \right )}+3570 C \,{\mathrm e}^{2 i \left (d x +c \right )}-91 A \,{\mathrm e}^{i \left (d x +c \right )}+1400 C \,{\mathrm e}^{i \left (d x +c \right )}-13 A +260 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(169\) |
| norman | \(\frac {\frac {C x}{a}+\frac {C x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {6 C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {4 C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (A -15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (A +C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (11 A -169 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}+\frac {\left (13 A -15 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}-\frac {\left (29 A -55 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{840 a d}+\frac {\left (47 A -1465 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}+\frac {\left (67 A -1145 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{120 a d}-\frac {\left (101 A +605 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{840 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{3}}\) | \(298\) |
1/840*(15*(A+C)*tan(1/2*d*x+1/2*c)^7+21*(-A-5*C)*tan(1/2*d*x+1/2*c)^5+35*( -A+11*C)*tan(1/2*d*x+1/2*c)^3+105*(A-15*C)*tan(1/2*d*x+1/2*c)+840*d*x*C)/a ^4/d
Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, C d x \cos \left (d x + c\right )^{4} + 420 \, C d x \cos \left (d x + c\right )^{3} + 630 \, C d x \cos \left (d x + c\right )^{2} + 420 \, C d x \cos \left (d x + c\right ) + 105 \, C d x + {\left (13 \, {\left (A - 20 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (13 \, A - 155 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (32 \, A - 535 \, C\right )} \cos \left (d x + c\right ) + 8 \, A - 160 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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 - 20*C)*cos(d*x + c)^3 + 4*(13*A - 155*C)*cos(d*x + c)^2 + (32*A - 535*C)*cos(d*x + c) + 8 *A - 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 = 2.88 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {C x}{a^{4}} + \frac {C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {11 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} - \frac {15 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{2}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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) + 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 + C*cos(c)**2)*cos(c)**2/(a*cos(c) + a)**4, Tru e))
Time = 0.37 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=-\frac {5 \, C {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac {A {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
-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)/d
Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (d x + c\right )} C}{a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 21 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
1/840*(840*(d*x + c)*C/a^4 + (15*A*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 - 105*C*a^24*ta n(1/2*d*x + 1/2*c)^5 - 35*A*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) - 1575*C*a^24*tan(1/2* d*x + 1/2*c))/a^28)/d
Time = 1.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx=\frac {C\,x}{a^4}+\frac {\left (\frac {13\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {52\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {13\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{210}+\frac {16\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {11\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{140}-\frac {5\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
(C*x)/a^4 + ((A*sin(c/2 + (d*x)/2))/56 + (C*sin(c/2 + (d*x)/2))/56 - cos(c /2 + (d*x)/2)^2*((11*A*sin(c/2 + (d*x)/2))/140 + (5*C*sin(c/2 + (d*x)/2))/ 28) + cos(c/2 + (d*x)/2)^6*((13*A*sin(c/2 + (d*x)/2))/105 - (52*C*sin(c/2 + (d*x)/2))/21) + cos(c/2 + (d*x)/2)^4*((13*A*sin(c/2 + (d*x)/2))/210 + (1 6*C*sin(c/2 + (d*x)/2))/21))/(a^4*d*cos(c/2 + (d*x)/2)^7)