Integrand size = 21, antiderivative size = 225 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
31/2*x/a^5-7664/315*sin(d*x+c)/a^5/d+31/2*cos(d*x+c)*sin(d*x+c)/a^5/d-1/9* cos(d*x+c)^6*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-17/63*cos(d*x+c)^5*sin(d*x+c) /a/d/(a+a*cos(d*x+c))^4-28/45*cos(d*x+c)^4*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c ))^3-577/315*cos(d*x+c)^3*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2-3832/315*cos (d*x+c)^2*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 8.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \csc ^{10}(c+d x) \sin ^9\left (\frac {1}{2} (c+d x)\right ) \left (984312+1035321 \cos (c+d x)-484476 \cos (2 (c+d x))-933309 \cos (3 (c+d x))-491576 \cos (4 (c+d x))-106807 \cos (5 (c+d x))-3780 \cos (6 (c+d x))+315 \cos (7 (c+d x))+9999360 \arcsin (\cos (c+d x)) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{1260 a^5 d} \]
-1/1260*(Cos[(c + d*x)/2]*Csc[c + d*x]^10*Sin[(c + d*x)/2]^9*(984312 + 103 5321*Cos[c + d*x] - 484476*Cos[2*(c + d*x)] - 933309*Cos[3*(c + d*x)] - 49 1576*Cos[4*(c + d*x)] - 106807*Cos[5*(c + d*x)] - 3780*Cos[6*(c + d*x)] + 315*Cos[7*(c + d*x)] + 9999360*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^8*Sqr t[Sin[c + d*x]^2]))/(a^5*d)
Time = 1.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3244, 3042, 3456, 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.
| \(\displaystyle \int \frac {\cos ^7(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^7}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {\cos ^5(c+d x) (6 a-11 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5 \left (6 a-11 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\int \frac {\cos ^4(c+d x) \left (85 a^2-111 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (85 a^2-111 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cos ^3(c+d x) \left (784 a^3-947 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (784 a^3-947 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {3 \cos ^2(c+d x) \left (1731 a^4-2101 a^4 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {577 a^3 \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (1731 a^4-2101 a^4 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{a^2}+\frac {577 a^3 \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (1731 a^4-2101 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{a^2}+\frac {577 a^3 \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {\int \cos (c+d x) \left (7664 a^5-9765 a^5 \cos (c+d x)\right )dx}{a^2}+\frac {3832 a^4 \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{a^2}+\frac {577 a^3 \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (7664 a^5-9765 a^5 \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {3832 a^4 \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}}{a^2}+\frac {577 a^3 \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle -\frac {\frac {\frac {196 a^2 \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {\frac {577 a^3 \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)^2}+\frac {\frac {3832 a^4 \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac {\frac {7664 a^5 \sin (c+d x)}{d}-\frac {9765 a^5 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {9765 a^5 x}{2}}{a^2}}{a^2}}{5 a^2}}{7 a^2}+\frac {17 a \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
-1/9*(Cos[c + d*x]^6*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^5) - ((17*a*Cos [c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((196*a^2*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((577*a^3*Cos[c + d*x ]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^2) + ((3832*a^4*Cos[c + d*x]^2*S in[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-9765*a^5*x)/2 + (7664*a^5*Sin[c + d*x])/d - (9765*a^5*Cos[c + d*x]*Sin[c + d*x])/(2*d))/a^2)/a^2)/(5*a^2) )/(7*a^2))/(9*a^2)
3.1.82.3.1 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[((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 = 1.00 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.44
| method | result | size |
| parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (6 d x +6 c \right )-\frac {854012 \cos \left (d x +c \right )}{63}-\frac {2250427 \cos \left (2 d x +2 c \right )}{315}-\frac {143054 \cos \left (3 d x +3 c \right )}{63}-\frac {113422 \cos \left (4 d x +4 c \right )}{315}-10 \cos \left (5 d x +5 c \right )-\frac {2627186}{315}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15872 d x}{1024 a^{5} d}\) | \(98\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(127\) |
| default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(127\) |
| risch | \(\frac {31 x}{2 a^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{5} d}+\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{5} d}-\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{5} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{5} d}-\frac {2 i \left (11025 \,{\mathrm e}^{8 i \left (d x +c \right )}+77175 \,{\mathrm e}^{7 i \left (d x +c \right )}+247695 \,{\mathrm e}^{6 i \left (d x +c \right )}+465255 \,{\mathrm e}^{5 i \left (d x +c \right )}+557109 \,{\mathrm e}^{4 i \left (d x +c \right )}+433881 \,{\mathrm e}^{3 i \left (d x +c \right )}+214929 \,{\mathrm e}^{2 i \left (d x +c \right )}+62001 \,{\mathrm e}^{i \left (d x +c \right )}+8114\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(192\) |
1/1024*(tan(1/2*d*x+1/2*c)*(cos(6*d*x+6*c)-854012/63*cos(d*x+c)-2250427/31 5*cos(2*d*x+2*c)-143054/63*cos(3*d*x+3*c)-113422/315*cos(4*d*x+4*c)-10*cos (5*d*x+5*c)-2627186/315)*sec(1/2*d*x+1/2*c)^8+15872*d*x)/a^5/d
Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x + {\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/630*(9765*d*x*cos(d*x + c)^5 + 48825*d*x*cos(d*x + c)^4 + 97650*d*x*cos( d*x + c)^3 + 97650*d*x*cos(d*x + c)^2 + 48825*d*x*cos(d*x + c) + 9765*d*x + (315*cos(d*x + c)^6 - 1575*cos(d*x + c)^5 - 28828*cos(d*x + c)^4 - 87440 *cos(d*x + c)^3 - 112119*cos(d*x + c)^2 - 66875*cos(d*x + c) - 15328)*sin( d*x + c))/(a^5*d*cos(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d* x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (214) = 428\).
Time = 19.70 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.61 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {78120 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {156240 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {78120 d x}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {35 \tan ^{13}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {380 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {2159 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {10152 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {82089 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {260820 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {155925 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{7}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((78120*d*x*tan(c/2 + d*x/2)**4/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5*d) + 156240*d*x*tan(c/2 + d *x/2)**2/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)* *2 + 5040*a**5*d) + 78120*d*x/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a** 5*d*tan(c/2 + d*x/2)**2 + 5040*a**5*d) - 35*tan(c/2 + d*x/2)**13/(5040*a** 5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5*d) + 380*tan(c/2 + d*x/2)**11/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d *tan(c/2 + d*x/2)**2 + 5040*a**5*d) - 2159*tan(c/2 + d*x/2)**9/(5040*a**5* d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5*d) + 10152*tan(c/2 + d*x/2)**7/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d* tan(c/2 + d*x/2)**2 + 5040*a**5*d) - 82089*tan(c/2 + d*x/2)**5/(5040*a**5* d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5*d) - 260820*tan(c/2 + d*x/2)**3/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d *tan(c/2 + d*x/2)**2 + 5040*a**5*d) - 155925*tan(c/2 + d*x/2)/(5040*a**5*d *tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5*d), Ne (d, 0)), (x*cos(c)**7/(a*cos(c) + a)**5, True))
Time = 0.34 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {156240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]
-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 11*sin(d*x + c)^3/(cos( d*x + c) + 1)^3)/(a^5 + 2*a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^5*si n(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c) + 1) - 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos( d*x + c) + 1)^5 - 450*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c )^9/(cos(d*x + c) + 1)^9)/a^5 - 156240*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d
Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {78120 \, {\left (d x + c\right )}}{a^{5}} - \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
1/5040*(78120*(d*x + c)/a^5 - 5040*(11*tan(1/2*d*x + 1/2*c)^3 + 9*tan(1/2* d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^5) - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1/2 *c)^5 - 15750*a^40*tan(1/2*d*x + 1/2*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2* c))/a^45)/d
Time = 15.43 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {35\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-590\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+4584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-23288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+129824\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+55440\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
-(35*sin(c/2 + (d*x)/2) - 590*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 45 84*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2) - 23288*cos(c/2 + (d*x)/2)^6*si n(c/2 + (d*x)/2) + 129824*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 55440* cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 10080*cos(c/2 + (d*x)/2)^12*sin (c/2 + (d*x)/2) - 78120*cos(c/2 + (d*x)/2)^9*(c + d*x))/(5040*a^5*d*cos(c/ 2 + (d*x)/2)^9)