Integrand size = 21, antiderivative size = 176 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {21 x}{2 a^4}-\frac {576 \sin (c+d x)}{35 a^4 d}+\frac {21 \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {43 \cos (c+d x) \sin (c+d x)}{35 a^4 d (1+\sec (c+d x))^2}-\frac {288 \cos (c+d x) \sin (c+d x)}{35 a^4 d (1+\sec (c+d x))}-\frac {\cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]
21/2*x/a^4-576/35*sin(d*x+c)/a^4/d+21/2*cos(d*x+c)*sin(d*x+c)/a^4/d-43/35* cos(d*x+c)*sin(d*x+c)/a^4/d/(1+sec(d*x+c))^2-288/35*cos(d*x+c)*sin(d*x+c)/ a^4/d/(1+sec(d*x+c))-1/7*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-2/5*co s(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^3
Time = 3.64 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (102900 d x \cos \left (\frac {d x}{2}\right )+102900 d x \cos \left (c+\frac {d x}{2}\right )+61740 d x \cos \left (c+\frac {3 d x}{2}\right )+61740 d x \cos \left (2 c+\frac {3 d x}{2}\right )+20580 d x \cos \left (2 c+\frac {5 d x}{2}\right )+20580 d x \cos \left (3 c+\frac {5 d x}{2}\right )+2940 d x \cos \left (3 c+\frac {7 d x}{2}\right )+2940 d x \cos \left (4 c+\frac {7 d x}{2}\right )-179830 \sin \left (\frac {d x}{2}\right )+128730 \sin \left (c+\frac {d x}{2}\right )-140826 \sin \left (c+\frac {3 d x}{2}\right )+44310 \sin \left (2 c+\frac {3 d x}{2}\right )-60487 \sin \left (2 c+\frac {5 d x}{2}\right )+1225 \sin \left (3 c+\frac {5 d x}{2}\right )-12001 \sin \left (3 c+\frac {7 d x}{2}\right )-3185 \sin \left (4 c+\frac {7 d x}{2}\right )-315 \sin \left (4 c+\frac {9 d x}{2}\right )-315 \sin \left (5 c+\frac {9 d x}{2}\right )+35 \sin \left (5 c+\frac {11 d x}{2}\right )+35 \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{35840 a^4 d} \]
(Sec[c/2]*Sec[(c + d*x)/2]^7*(102900*d*x*Cos[(d*x)/2] + 102900*d*x*Cos[c + (d*x)/2] + 61740*d*x*Cos[c + (3*d*x)/2] + 61740*d*x*Cos[2*c + (3*d*x)/2] + 20580*d*x*Cos[2*c + (5*d*x)/2] + 20580*d*x*Cos[3*c + (5*d*x)/2] + 2940*d *x*Cos[3*c + (7*d*x)/2] + 2940*d*x*Cos[4*c + (7*d*x)/2] - 179830*Sin[(d*x) /2] + 128730*Sin[c + (d*x)/2] - 140826*Sin[c + (3*d*x)/2] + 44310*Sin[2*c + (3*d*x)/2] - 60487*Sin[2*c + (5*d*x)/2] + 1225*Sin[3*c + (5*d*x)/2] - 12 001*Sin[3*c + (7*d*x)/2] - 3185*Sin[4*c + (7*d*x)/2] - 315*Sin[4*c + (9*d* x)/2] - 315*Sin[5*c + (9*d*x)/2] + 35*Sin[5*c + (11*d*x)/2] + 35*Sin[6*c + (11*d*x)/2]))/(35840*a^4*d)
Time = 1.22 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.10, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 4304, 25, 3042, 4508, 3042, 4508, 27, 3042, 4508, 3042, 4274, 3042, 3115, 24, 3117}
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.
| \(\displaystyle \int \frac {\cos ^2(c+d x)}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 4304 |
| \(\displaystyle -\frac {\int -\frac {\cos ^2(c+d x) (9 a-5 a \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (9 a-5 a \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {9 a-5 a \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^2(c+d x) \left (73 a^2-56 a^2 \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {73 a^2-56 a^2 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {9 \cos ^2(c+d x) \left (53 a^3-43 a^3 \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\cos ^2(c+d x) \left (53 a^3-43 a^3 \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {53 a^3-43 a^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \cos ^2(c+d x) \left (245 a^4-192 a^4 \sec (c+d x)\right )dx}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {245 a^4-192 a^4 \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {245 a^4 \int \cos ^2(c+d x)dx-192 a^4 \int \cos (c+d x)dx}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {245 a^4 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-192 a^4 \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {245 a^4 \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-192 a^4 \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {245 a^4 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-192 a^4 \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {245 a^4 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {192 a^4 \sin (c+d x)}{d}}{a^2}-\frac {96 a^3 \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}\right )}{a^2}-\frac {43 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {14 a \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
-1/7*(Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^4) + ((-14*a*Cos[ c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((-43*Cos[c + d*x]*S in[c + d*x])/(d*(1 + Sec[c + d*x])^2) + (3*((-96*a^3*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((-192*a^4*Sin[c + d*x])/d + 245*a^4*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/a^2))/a^2)/(5*a^2))/(7*a^2)
3.1.79.3.1 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[((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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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_), x_Symbol] :> Simp[(-Cot[e + f*x])*(a + b*Csc[e + f*x])^m*((d*Csc [e + f*x])^n/(f*(2*m + 1))), x] + Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a*(2*m + n + 1) - b*(m + n + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ [m, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m])
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]
Time = 0.57 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.50
| method | result | size |
| parallelrisch | \(\frac {-85762 \left (\cos \left (d x +c \right )+\frac {18752 \cos \left (2 d x +2 c \right )}{42881}+\frac {7873 \cos \left (3 d x +3 c \right )}{85762}+\frac {140 \cos \left (4 d x +4 c \right )}{42881}-\frac {35 \cos \left (5 d x +5 c \right )}{85762}+\frac {27828}{42881}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+94080 d x}{8960 a^{4} d}\) | \(88\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+168 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(114\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-72 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+168 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(114\) |
| risch | \(\frac {21 x}{2 a^{4}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (700 \,{\mathrm e}^{6 i \left (d x +c \right )}+3675 \,{\mathrm e}^{5 i \left (d x +c \right )}+8505 \,{\mathrm e}^{4 i \left (d x +c \right )}+10780 \,{\mathrm e}^{3 i \left (d x +c \right )}+7896 \,{\mathrm e}^{2 i \left (d x +c \right )}+3157 \,{\mathrm e}^{i \left (d x +c \right )}+551\right )}{35 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(170\) |
| norman | \(\frac {\frac {21 x}{2 a}-\frac {167 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {281 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a d}-\frac {217 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {167 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{140 a d}-\frac {53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{280 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{56 a d}+\frac {21 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {21 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a^{3}}\) | \(173\) |
1/8960*(-85762*(cos(d*x+c)+18752/42881*cos(2*d*x+2*c)+7873/85762*cos(3*d*x +3*c)+140/42881*cos(4*d*x+4*c)-35/85762*cos(5*d*x+5*c)+27828/42881)*tan(1/ 2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6+94080*d*x)/a^4/d
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {735 \, d x \cos \left (d x + c\right )^{4} + 2940 \, d x \cos \left (d x + c\right )^{3} + 4410 \, d x \cos \left (d x + c\right )^{2} + 2940 \, d x \cos \left (d x + c\right ) + 735 \, d x + {\left (35 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 2012 \, \cos \left (d x + c\right )^{3} - 4548 \, \cos \left (d x + c\right )^{2} - 3873 \, \cos \left (d x + c\right ) - 1152\right )} \sin \left (d x + c\right )}{70 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
1/70*(735*d*x*cos(d*x + c)^4 + 2940*d*x*cos(d*x + c)^3 + 4410*d*x*cos(d*x + c)^2 + 2940*d*x*cos(d*x + c) + 735*d*x + (35*cos(d*x + c)^5 - 140*cos(d* x + c)^4 - 2012*cos(d*x + c)^3 - 4548*cos(d*x + c)^2 - 3873*cos(d*x + c) - 1152)*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)
\[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Integral(cos(c + d*x)**2/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x)/a**4
Time = 0.34 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{280 \, d} \]
-1/280*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^4 + 2*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4*sin(d *x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) - 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*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 - 5880*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4)/d
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.73 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {2940 \, {\left (d x + c\right )}}{a^{4}} - \frac {280 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {5 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 455 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3885 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{280 \, d} \]
1/280*(2940*(d*x + c)/a^4 - 280*(9*tan(1/2*d*x + 1/2*c)^3 + 7*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^4) + (5*a^24*tan(1/2*d*x + 1/2 *c)^7 - 63*a^24*tan(1/2*d*x + 1/2*c)^5 + 455*a^24*tan(1/2*d*x + 1/2*c)^3 - 3885*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 13.61 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+596\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-4408\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2940\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )}{280\,a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
(5*sin(c/2 + (d*x)/2) - 78*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 596*c os(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2) - 4408*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) - 2520*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 560*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) + 2940*cos(c/2 + (d*x)/2)^7*(c + d*x))/(28 0*a^4*d*cos(c/2 + (d*x)/2)^7)