Integrand size = 21, antiderivative size = 184 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {21 x}{2 a^4}-\frac {576 \sin (c+d x)}{35 a^4 d}+\frac {21 \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {43 \cos ^3(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))^2}-\frac {288 \cos ^2(c+d x) \sin (c+d x)}{35 a^4 d (1+\cos (c+d x))}-\frac {\cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3} \] Output:
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)^3*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2-288/35*cos(d*x+c)^2*sin(d*x +c)/a^4/d/(1+cos(d*x+c))-1/7*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^4- 2/5*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(993\) vs. \(2(184)=368\).
Time = 8.38 (sec) , antiderivative size = 993, normalized size of antiderivative = 5.40 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4,x]
Output:
-((Sin[c + d*x]*(-1/2*(Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^5)/(1 + Cos[c + d*x])^(7/2) + (-4*(-3*ArcSin[Cos[c + d*x]] - (6*Sqrt[1 - Cos[c + d*x]])/( 5*(1 + Cos[c + d*x])^(5/2)) - (Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^3)/(1 + Cos[c + d*x])^(5/2) + (21*Sqrt[1 - Cos[c + d*x]])/(5*(1 + Cos[c + d*x])^( 3/2)) - (39*Sqrt[1 - Cos[c + d*x]])/(5*Sqrt[1 + Cos[c + d*x]])) + 9*(-32/( 105*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7/2)) - (3*Cos[c + d*x])/ (28*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7/2)) + (16*Cos[c + d*x]^ 2)/(15*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7/2)) + Cos[c + d*x]^3 /(4*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7/2)) + (32*Sqrt[1 - Cos[ c + d*x]]*Cos[c + d*x]^3)/(7*(1 + Cos[c + d*x])^(7/2)) - (4*Cos[c + d*x]^4 )/(3*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7/2)) + (8*Cos[c + d*x]^ 4)/(Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^(7/2)) - (8*Cos[c + d*x]^5)/ (5*(1 - Cos[c + d*x])^(3/2)*(1 + Cos[c + d*x])^(7/2)) + (4*Cos[c + d*x]^6) /(5*(1 - Cos[c + d*x])^(5/2)*(1 + Cos[c + d*x])^(7/2)) + (2*Cos[c + d*x]^7 )/(7*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(7/2)) + 3/(14*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d*x])^(5/2)) + (96*Sqrt[1 - Cos[c + d*x]])/(35 *(1 + Cos[c + d*x])^(5/2)) + (9*Cos[c + d*x])/(70*(1 - Cos[c + d*x])^(5/2) *(1 + Cos[c + d*x])^(5/2)) - 3/(28*(1 - Cos[c + d*x])^(7/2)*(1 + Cos[c + d *x])^(3/2)) - 3/(28*(1 - Cos[c + d*x])^(5/2)*(1 + Cos[c + d*x])^(3/2)) - ( 256*Sqrt[1 - Cos[c + d*x]])/(35*(1 + Cos[c + d*x])^(3/2)) + Cos[c + d*x...
Time = 0.98 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 3244, 3042, 3456, 3042, 3456, 27, 3042, 3456, 3042, 3213}
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.
| \(\displaystyle \int \frac {\cos ^6(c+d x)}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^6}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {\cos ^4(c+d x) (5 a-9 a \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (5 a-9 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\int \frac {\cos ^3(c+d x) \left (56 a^2-73 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (56 a^2-73 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {9 \cos ^2(c+d x) \left (43 a^3-53 a^3 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {43 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {\cos ^2(c+d x) \left (43 a^3-53 a^3 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{a^2}+\frac {43 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (43 a^3-53 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{a^2}+\frac {43 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\int \cos (c+d x) \left (192 a^4-245 a^4 \cos (c+d x)\right )dx}{a^2}+\frac {96 a^3 \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{a^2}+\frac {43 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (192 a^4-245 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {96 a^3 \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{a^2}+\frac {43 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {96 a^3 \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac {\frac {192 a^4 \sin (c+d x)}{d}-\frac {245 a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {245 a^4 x}{2}}{a^2}\right )}{a^2}+\frac {43 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {14 a \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
Input:
Int[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4,x]
Output:
-1/7*(Cos[c + d*x]^5*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) - ((14*a*Cos [c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((43*Cos[c + d*x] ^3*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^2) + (3*((96*a^3*Cos[c + d*x]^2*Sin [c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-245*a^4*x)/2 + (192*a^4*Sin[c + d *x])/d - (245*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^2))/a^2)/(5*a^2))/(7 *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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(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 - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 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])
Time = 1.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.49
| method | result | size |
| parallelrisch | \(\frac {94080 d x -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (55656-35 \cos \left (5 d x +5 c \right )+280 \cos \left (4 d x +4 c \right )+7873 \cos \left (3 d x +3 c \right )+37504 \cos \left (2 d x +2 c \right )+85762 \cos \left (d x +c \right )\right )}{8960 a^{4} d}\) | \(90\) |
| 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 | \(\frac {21 x}{2 a^{4}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (700 \,{\mathrm e}^{6 i \left (d x +c \right )}+3675 \,{\mathrm e}^{5 i \left (d x +c \right )}+8505 \,{\mathrm e}^{4 i \left (d x +c \right )}+10780 \,{\mathrm e}^{3 i \left (d x +c \right )}+7896 \,{\mathrm e}^{2 i \left (d x +c \right )}+3157 \,{\mathrm e}^{i \left (d x +c \right )}+551\right )}{35 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(170\) |
Input:
int(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/8960*(94080*d*x-tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6*(55656-35*cos(5* d*x+5*c)+280*cos(4*d*x+4*c)+7873*cos(3*d*x+3*c)+37504*cos(2*d*x+2*c)+85762 *cos(d*x+c)))/a^4/d
Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {735 \, d x \cos \left (d x + c\right )^{4} + 2940 \, d x \cos \left (d x + c\right )^{3} + 4410 \, d x \cos \left (d x + c\right )^{2} + 2940 \, d x \cos \left (d x + c\right ) + 735 \, d x + {\left (35 \, \cos \left (d x + c\right )^{5} - 140 \, \cos \left (d x + c\right )^{4} - 2012 \, \cos \left (d x + c\right )^{3} - 4548 \, \cos \left (d x + c\right )^{2} - 3873 \, \cos \left (d x + c\right ) - 1152\right )} \sin \left (d x + c\right )}{70 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:
integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x, algorithm="fricas")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (175) = 350\).
Time = 7.27 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.88 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {2940 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {5880 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {2940 d x}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {5 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {53 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} + \frac {334 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {3038 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {9835 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} - \frac {5845 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{280 a^{4} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 560 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 280 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{6}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6/(a+a*cos(d*x+c))**4,x)
Output:
Piecewise((2940*d*x*tan(c/2 + d*x/2)**4/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) + 5880*d*x*tan(c/2 + d*x/2)** 2/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a **4*d) + 2940*d*x/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d *x/2)**2 + 280*a**4*d) + 5*tan(c/2 + d*x/2)**11/(280*a**4*d*tan(c/2 + d*x/ 2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 53*tan(c/2 + d*x/2) **9/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280 *a**4*d) + 334*tan(c/2 + d*x/2)**7/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a **4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 3038*tan(c/2 + d*x/2)**5/(280*a* *4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d) - 9835*tan(c/2 + d*x/2)**3/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan( c/2 + d*x/2)**2 + 280*a**4*d) - 5845*tan(c/2 + d*x/2)/(280*a**4*d*tan(c/2 + d*x/2)**4 + 560*a**4*d*tan(c/2 + d*x/2)**2 + 280*a**4*d), Ne(d, 0)), (x* cos(c)**6/(a*cos(c) + a)**4, True))
Time = 0.11 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{280 \, d} \] Input:
integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x, algorithm="maxima")
Output:
-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 = 0.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {2940 \, {\left (d x + c\right )}}{a^{4}} - \frac {280 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {5 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 455 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3885 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{280 \, d} \] Input:
integrate(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x, algorithm="giac")
Output:
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 = 41.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+596\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-4408\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2940\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )}{280\,a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int(cos(c + d*x)^6/(a + a*cos(c + d*x))^4,x)
Output:
(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)
Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-53 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+334 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-3038 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2940 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} d x -9835 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+5880 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d x -5845 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2940 d x}{280 a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(cos(d*x+c)^6/(a+a*cos(d*x+c))^4,x)
Output:
(5*tan((c + d*x)/2)**11 - 53*tan((c + d*x)/2)**9 + 334*tan((c + d*x)/2)**7 - 3038*tan((c + d*x)/2)**5 + 2940*tan((c + d*x)/2)**4*d*x - 9835*tan((c + d*x)/2)**3 + 5880*tan((c + d*x)/2)**2*d*x - 5845*tan((c + d*x)/2) + 2940* d*x)/(280*a**4*d*(tan((c + d*x)/2)**4 + 2*tan((c + d*x)/2)**2 + 1))