Integrand size = 21, antiderivative size = 191 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {5 x}{a^5}+\frac {181 \sin (c+d x)}{63 a^5 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {5 \cos ^4(c+d x) \sin (c+d x)}{21 a d (a+a \cos (c+d x))^4}-\frac {29 \cos ^3(c+d x) \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}-\frac {67 \cos ^2(c+d x) \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac {5 \sin (c+d x)}{d \left (a^5+a^5 \cos (c+d x)\right )} \]
-5*x/a^5+181/63*sin(d*x+c)/a^5/d-1/9*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d* x+c))^5-5/21*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4-29/63*cos(d*x+ c)^3*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3-67/63*cos(d*x+c)^2*sin(d*x+c)/a^3 /d/(a+a*cos(d*x+c))^2+5*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 7.98 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \sin ^9\left (\frac {1}{2} (c+d x)\right ) \left (161280 \arcsin (\cos (c+d x)) \cos ^{10}\left (\frac {1}{2} (c+d x)\right )+(42676+69350 \cos (c+d x)+36632 \cos (2 (c+d x))+11675 \cos (3 (c+d x))+1892 \cos (4 (c+d x))+63 \cos (5 (c+d x))) \sqrt {\sin ^2(c+d x)}\right )}{63 a^5 d (-1+\cos (c+d x))^4 (1+\cos (c+d x))^5 \sqrt {\sin ^2(c+d x)}} \]
(2*Cos[(c + d*x)/2]*Sin[(c + d*x)/2]^9*(161280*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^10 + (42676 + 69350*Cos[c + d*x] + 36632*Cos[2*(c + d*x)] + 116 75*Cos[3*(c + d*x)] + 1892*Cos[4*(c + d*x)] + 63*Cos[5*(c + d*x)])*Sqrt[Si n[c + d*x]^2]))/(63*a^5*d*(-1 + Cos[c + d*x])^4*(1 + Cos[c + d*x])^5*Sqrt[ Sin[c + d*x]^2])
Time = 1.50 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.15, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3244, 27, 3042, 3456, 3042, 3456, 27, 3042, 3456, 3042, 3447, 3042, 3502, 27, 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.
| \(\displaystyle \int \frac {\cos ^6(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^6}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {5 \cos ^4(c+d x) (a-2 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \int \frac {\cos ^4(c+d x) (a-2 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {5 \left (\frac {\int \frac {\cos ^3(c+d x) \left (12 a^2-17 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (12 a^2-17 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {\int \frac {3 \cos ^2(c+d x) \left (29 a^3-38 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \int \frac {\cos ^2(c+d x) \left (29 a^3-38 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (29 a^3-38 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {\int \frac {\cos (c+d x) \left (134 a^4-181 a^4 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (134 a^4-181 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {\int \frac {134 a^4 \cos (c+d x)-181 a^4 \cos ^2(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {\int \frac {134 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )-181 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {315 a^5 \cos (c+d x)}{\cos (c+d x) a+a}dx}{a}-\frac {181 a^3 \sin (c+d x)}{d}}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {315 a^4 \int \frac {\cos (c+d x)}{\cos (c+d x) a+a}dx-\frac {181 a^3 \sin (c+d x)}{d}}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {315 a^4 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {181 a^3 \sin (c+d x)}{d}}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {315 a^4 \left (\frac {x}{a}-\int \frac {1}{\cos (c+d x) a+a}dx\right )-\frac {181 a^3 \sin (c+d x)}{d}}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {3 \left (\frac {315 a^4 \left (\frac {x}{a}-\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )-\frac {181 a^3 \sin (c+d x)}{d}}{3 a^2}+\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a^2}+\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {5 \left (\frac {\frac {29 a^2 \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {3 \left (\frac {67 a^3 \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}+\frac {315 a^4 \left (\frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)}\right )-\frac {181 a^3 \sin (c+d x)}{d}}{3 a^2}\right )}{5 a^2}}{7 a^2}+\frac {3 a \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a^2}-\frac {\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
-1/9*(Cos[c + d*x]^5*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^5) - (5*((3*a*C os[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((29*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (3*((67*a^3*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((-181*a^3*Sin[c + d*x ])/d + 315*a^4*(x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))))/(3*a^2)))/(5 *a^2))/(7*a^2)))/(9*a^2)
3.1.83.3.1 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[(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_)*((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_)]), 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[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 0.74 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.46
| method | result | size |
| parallelrisch | \(\frac {\frac {34675 \left (\cos \left (d x +c \right )+\frac {964 \cos \left (2 d x +2 c \right )}{1825}+\frac {467 \cos \left (3 d x +3 c \right )}{2774}+\frac {946 \cos \left (4 d x +4 c \right )}{34675}+\frac {63 \cos \left (5 d x +5 c \right )}{69350}+\frac {21338}{34675}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8064}-5 d x}{a^{5} d}\) | \(88\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-160 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(111\) |
| default | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-160 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(111\) |
| risch | \(-\frac {5 x}{a^{5}}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{5} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{5} d}+\frac {2 i \left (945 \,{\mathrm e}^{8 i \left (d x +c \right )}+6300 \,{\mathrm e}^{7 i \left (d x +c \right )}+19740 \,{\mathrm e}^{6 i \left (d x +c \right )}+36414 \,{\mathrm e}^{5 i \left (d x +c \right )}+43092 \,{\mathrm e}^{4 i \left (d x +c \right )}+33264 \,{\mathrm e}^{3 i \left (d x +c \right )}+16416 \,{\mathrm e}^{2 i \left (d x +c \right )}+4734 \,{\mathrm e}^{i \left (d x +c \right )}+631\right )}{63 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(156\) |
5/8064*(6935*(cos(d*x+c)+964/1825*cos(2*d*x+2*c)+467/2774*cos(3*d*x+3*c)+9 46/34675*cos(4*d*x+4*c)+63/69350*cos(5*d*x+5*c)+21338/34675)*tan(1/2*d*x+1 /2*c)*sec(1/2*d*x+1/2*c)^8-8064*d*x)/a^5/d
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (63 \, \cos \left (d x + c\right )^{5} + 946 \, \cos \left (d x + c\right )^{4} + 2840 \, \cos \left (d x + c\right )^{3} + 3633 \, \cos \left (d x + c\right )^{2} + 2165 \, \cos \left (d x + c\right ) + 496\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
-1/63*(315*d*x*cos(d*x + c)^5 + 1575*d*x*cos(d*x + c)^4 + 3150*d*x*cos(d*x + c)^3 + 3150*d*x*cos(d*x + c)^2 + 1575*d*x*cos(d*x + c) + 315*d*x - (63* cos(d*x + c)^5 + 946*cos(d*x + c)^4 + 2840*cos(d*x + c)^3 + 3633*cos(d*x + c)^2 + 2165*cos(d*x + c) + 496)*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)
Time = 11.54 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} - \frac {5040 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} - \frac {5040 d x}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {7 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} - \frac {65 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {306 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} - \frac {1134 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {6615 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} + \frac {10143 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1008 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1008 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{6}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((-5040*d*x*tan(c/2 + d*x/2)**2/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) - 5040*d*x/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 7*tan(c/2 + d*x/2)**11/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) - 65*tan(c/2 + d*x/2)**9/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 306*tan(c/2 + d*x/2)**7/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) - 1134*tan(c/2 + d*x/2)**5/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 6615*tan(c/2 + d*x/2)**3/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d) + 10143*tan(c/2 + d*x/2)/(1008*a**5*d*tan(c/2 + d*x/2)**2 + 1008*a**5*d), Ne(d, 0)), (x*cos(c)**6/(a*cos(c) + a)**5, True))
Time = 0.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {2016 \, \sin \left (d x + c\right )}{{\left (a^{5} + \frac {a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {8127 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1512 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {72 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{1008 \, d} \]
1/1008*(2016*sin(d*x + c)/((a^5 + a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2) *(cos(d*x + c) + 1)) + (8127*sin(d*x + c)/(cos(d*x + c) + 1) - 1512*sin(d* x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 72*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 7*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^5 - 10080*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d
Time = 0.43 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {2016 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{5}} - \frac {7 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 72 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1512 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8127 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{1008 \, d} \]
-1/1008*(5040*(d*x + c)/a^5 - 2016*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/ 2*c)^2 + 1)*a^5) - (7*a^40*tan(1/2*d*x + 1/2*c)^9 - 72*a^40*tan(1/2*d*x + 1/2*c)^7 + 378*a^40*tan(1/2*d*x + 1/2*c)^5 - 1512*a^40*tan(1/2*d*x + 1/2*c )^3 + 8127*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d
Time = 15.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-100\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+636\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2512\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+10096\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2016\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{1008\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
(7*sin(c/2 + (d*x)/2) - 100*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 636* cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2) - 2512*cos(c/2 + (d*x)/2)^6*sin(c/ 2 + (d*x)/2) + 10096*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 2016*cos(c/ 2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 5040*cos(c/2 + (d*x)/2)^9*(c + d*x))/ (1008*a^5*d*cos(c/2 + (d*x)/2)^9)