Integrand size = 21, antiderivative size = 168 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\cos ^4(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {13 \cos ^3(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {34 \cos ^2(c+d x) \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {173 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {661 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
x/a^5-1/9*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-13/63*cos(d*x+c)^3* sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4-34/105*cos(d*x+c)^2*sin(d*x+c)/a^2/d/(a+ a*cos(d*x+c))^3+173/315*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2-661/315*sin(d* x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 7.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {8 \cos \left (\frac {1}{2} (c+d x)\right ) \csc ^{10}(c+d x) \sin ^{11}\left (\frac {1}{2} (c+d x)\right ) \left (80640 \arcsin (\cos (c+d x)) \cos ^{10}\left (\frac {1}{2} (c+d x)\right )+(20689+33440 \cos (c+d x)+17648 \cos (2 (c+d x))+5480 \cos (3 (c+d x))+863 \cos (4 (c+d x))) \sqrt {\sin ^2(c+d x)}\right )}{315 a^5 d \sqrt {\sin ^2(c+d x)}} \]
(-8*Cos[(c + d*x)/2]*Csc[c + d*x]^10*Sin[(c + d*x)/2]^11*(80640*ArcSin[Cos [c + d*x]]*Cos[(c + d*x)/2]^10 + (20689 + 33440*Cos[c + d*x] + 17648*Cos[2 *(c + d*x)] + 5480*Cos[3*(c + d*x)] + 863*Cos[4*(c + d*x)])*Sqrt[Sin[c + d *x]^2]))/(315*a^5*d*Sqrt[Sin[c + d*x]^2])
Time = 1.25 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.17, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3244, 3042, 3456, 27, 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.
\(\displaystyle \int \frac {\cos ^5(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
\(\Big \downarrow \) 3244 |
\(\displaystyle -\frac {\int \frac {\cos ^3(c+d x) (4 a-9 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (4 a-9 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3456 |
\(\displaystyle -\frac {\frac {\int \frac {3 \cos ^2(c+d x) \left (13 a^2-21 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {3 \int \frac {\cos ^2(c+d x) \left (13 a^2-21 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (13 a^2-21 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3456 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {\cos (c+d x) \left (68 a^3-105 a^3 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (68 a^3-105 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3447 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {68 a^3 \cos (c+d x)-105 a^3 \cos ^2(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {68 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )-105 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3498 |
\(\displaystyle -\frac {\frac {3 \left (\frac {-\frac {\int -\frac {346 a^4-315 a^4 \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\int \frac {346 a^4-315 a^4 \cos (c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {\int \frac {346 a^4-315 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {661 a^4 \int \frac {1}{\cos (c+d x) a+a}dx-315 a^3 x}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {3 \left (\frac {\frac {661 a^4 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-315 a^3 x}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}+\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle -\frac {\frac {3 \left (\frac {34 a^2 \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {\frac {\frac {661 a^4 \sin (c+d x)}{d (a \cos (c+d x)+a)}-315 a^3 x}{3 a^2}-\frac {173 a^3 \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}\right )}{7 a^2}+\frac {13 a \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
-1/9*(Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^5) - ((13*a*Cos [c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (3*((34*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((-173*a^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (-315*a^3*x + (661*a^4*Sin[c + d*x])/ (d*(a + a*Cos[c + d*x])))/(3*a^2))/(5*a^2)))/(7*a^2))/(9*a^2)
3.1.84.3.1 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[(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]
Time = 0.71 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46
method | result | size |
parallelrisch | \(\frac {-35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+270 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2730 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5040 d x -9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5040 a^{5} d}\) | \(77\) |
derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(85\) |
default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {6 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {26 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(85\) |
risch | \(\frac {x}{a^{5}}-\frac {2 i \left (1575 \,{\mathrm e}^{8 i \left (d x +c \right )}+9450 \,{\mathrm e}^{7 i \left (d x +c \right )}+28350 \,{\mathrm e}^{6 i \left (d x +c \right )}+50400 \,{\mathrm e}^{5 i \left (d x +c \right )}+58338 \,{\mathrm e}^{4 i \left (d x +c \right )}+44142 \,{\mathrm e}^{3 i \left (d x +c \right )}+21618 \,{\mathrm e}^{2 i \left (d x +c \right )}+6192 \,{\mathrm e}^{i \left (d x +c \right )}+863\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(119\) |
1/5040*(-35*tan(1/2*d*x+1/2*c)^9+270*tan(1/2*d*x+1/2*c)^7-1008*tan(1/2*d*x +1/2*c)^5+2730*tan(1/2*d*x+1/2*c)^3+5040*d*x-9765*tan(1/2*d*x+1/2*c))/a^5/ d
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.12 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {315 \, d x \cos \left (d x + c\right )^{5} + 1575 \, d x \cos \left (d x + c\right )^{4} + 3150 \, d x \cos \left (d x + c\right )^{3} + 3150 \, d x \cos \left (d x + c\right )^{2} + 1575 \, d x \cos \left (d x + c\right ) + 315 \, d x - {\left (863 \, \cos \left (d x + c\right )^{4} + 2740 \, \cos \left (d x + c\right )^{3} + 3549 \, \cos \left (d x + c\right )^{2} + 2125 \, \cos \left (d x + c\right ) + 488\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/315*(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 - (863 *cos(d*x + c)^4 + 2740*cos(d*x + c)^3 + 3549*cos(d*x + c)^2 + 2125*cos(d*x + c) + 488)*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 = 7.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {x}{a^{5}} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {3 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5 a^{5} d} + \frac {13 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} - \frac {31 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((x/a**5 - tan(c/2 + d*x/2)**9/(144*a**5*d) + 3*tan(c/2 + d*x/2)* *7/(56*a**5*d) - tan(c/2 + d*x/2)**5/(5*a**5*d) + 13*tan(c/2 + d*x/2)**3/( 24*a**5*d) - 31*tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x*cos(c)**5/(a*c os(c) + a)**5, True))
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {\frac {9765 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2730 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {10080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]
-1/5040*((9765*sin(d*x + c)/(cos(d*x + c) + 1) - 2730*sin(d*x + c)^3/(cos( d*x + c) + 1)^3 + 1008*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 270*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 - 10080*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^5)/d
Time = 0.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {5040 \, {\left (d x + c\right )}}{a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 270 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1008 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2730 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9765 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
1/5040*(5040*(d*x + c)/a^5 - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 270*a^40*ta n(1/2*d*x + 1/2*c)^7 + 1008*a^40*tan(1/2*d*x + 1/2*c)^5 - 2730*a^40*tan(1/ 2*d*x + 1/2*c)^3 + 9765*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d
Time = 14.85 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {x}{a^5}-\frac {\frac {863\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{315}-\frac {356\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{315}+\frac {169\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{420}-\frac {41\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{504}+\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{144}}{a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]