Integrand size = 21, antiderivative size = 150 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {4 x}{a^4}+\frac {244 \sin (c+d x)}{105 a^4 d}-\frac {88 \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {4 \sin (c+d x)}{a^4 d (1+\cos (c+d x))}-\frac {\cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {12 \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
-4*x/a^4+244/105*sin(d*x+c)/a^4/d-88/105*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+ cos(d*x+c))^2+4*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*cos(d*x+c)^4*sin(d*x+c )/d/(a+a*cos(d*x+c))^4-12/35*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^ 3
Time = 1.66 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sin ^5\left (\frac {1}{2} (c+d x)\right ) \left (7350+6678 \cos (c+d x)-5432 \cos (2 (c+d x))-6333 \cos (3 (c+d x))-2158 \cos (4 (c+d x))-105 \cos (5 (c+d x))+53760 \arcsin (\cos (c+d x)) \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{210 a^4 d (-1+\cos (c+d x))^3 (1+\cos (c+d x))^4} \]
-1/210*(Cos[(c + d*x)/2]*Sin[(c + d*x)/2]^5*(7350 + 6678*Cos[c + d*x] - 54 32*Cos[2*(c + d*x)] - 6333*Cos[3*(c + d*x)] - 2158*Cos[4*(c + d*x)] - 105* Cos[5*(c + d*x)] + 53760*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^6*Sqrt[Sin[ c + d*x]^2]))/(a^4*d*(-1 + Cos[c + d*x])^3*(1 + Cos[c + d*x])^4)
Time = 1.19 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.14, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3244, 27, 3042, 3456, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {4 \cos ^3(c+d x) (a-2 a \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \int \frac {\cos ^3(c+d x) (a-2 a \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {4 \left (\frac {\int \frac {\cos ^2(c+d x) \left (9 a^2-13 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (9 a^2-13 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {\int \frac {\cos (c+d x) \left (44 a^3-61 a^3 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (44 a^3-61 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {\int \frac {44 a^3 \cos (c+d x)-61 a^3 \cos ^2(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {\int \frac {44 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )-61 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {\frac {\int \frac {105 a^4 \cos (c+d x)}{\cos (c+d x) a+a}dx}{a}-\frac {61 a^2 \sin (c+d x)}{d}}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {105 a^3 \int \frac {\cos (c+d x)}{\cos (c+d x) a+a}dx-\frac {61 a^2 \sin (c+d x)}{d}}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {105 a^3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {61 a^2 \sin (c+d x)}{d}}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {105 a^3 \left (\frac {x}{a}-\int \frac {1}{\cos (c+d x) a+a}dx\right )-\frac {61 a^2 \sin (c+d x)}{d}}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {105 a^3 \left (\frac {x}{a}-\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )-\frac {61 a^2 \sin (c+d x)}{d}}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {4 \left (\frac {\frac {105 a^3 \left (\frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)}\right )-\frac {61 a^2 \sin (c+d x)}{d}}{3 a^2}+\frac {22 \sin (c+d x) \cos ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {3 a \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*(Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^4) - (4*((3*a*C os[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((22*Cos[c + d* x]^2*Sin[c + d*x])/(3*d*(1 + Cos[c + d*x])^2) + ((-61*a^2*Sin[c + d*x])/d + 105*a^3*(x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))))/(3*a^2))/(5*a^2)) )/(7*a^2)
3.1.73.3.1 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 = 0.83 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.51
| method | result | size |
| parallelrisch | \(\frac {781 \left (\cos \left (d x +c \right )+\frac {2741 \cos \left (2 d x +2 c \right )}{6248}+\frac {74 \cos \left (3 d x +3 c \right )}{781}+\frac {105 \cos \left (4 d x +4 c \right )}{24992}+\frac {16171}{24992}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-840 d x}{210 a^{4} d}\) | \(77\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-64 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(98\) |
| default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-64 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(98\) |
| risch | \(-\frac {4 x}{a^{4}}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{4}}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{4}}+\frac {4 i \left (525 \,{\mathrm e}^{6 i \left (d x +c \right )}+2625 \,{\mathrm e}^{5 i \left (d x +c \right )}+5950 \,{\mathrm e}^{4 i \left (d x +c \right )}+7420 \,{\mathrm e}^{3 i \left (d x +c \right )}+5397 \,{\mathrm e}^{2 i \left (d x +c \right )}+2149 \,{\mathrm e}^{i \left (d x +c \right )}+382\right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(134\) |
| norman | \(\frac {-\frac {4 x}{a}-\frac {\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )}{56 a d}+\frac {65 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {113 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {2059 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d a}+\frac {1271 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}+\frac {2075 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 d a}+\frac {61 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}-\frac {11 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{42 d a}+\frac {3 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}-\frac {20 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {20 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{3}}\) | \(281\) |
1/210*(781*(cos(d*x+c)+2741/6248*cos(2*d*x+2*c)+74/781*cos(3*d*x+3*c)+105/ 24992*cos(4*d*x+4*c)+16171/24992)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6- 840*d*x)/a^4/d
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {420 \, d x \cos \left (d x + c\right )^{4} + 1680 \, d x \cos \left (d x + c\right )^{3} + 2520 \, d x \cos \left (d x + c\right )^{2} + 1680 \, d x \cos \left (d x + c\right ) + 420 \, d x - {\left (105 \, \cos \left (d x + c\right )^{4} + 1184 \, \cos \left (d x + c\right )^{3} + 2636 \, \cos \left (d x + c\right )^{2} + 2236 \, \cos \left (d x + c\right ) + 664\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
-1/105*(420*d*x*cos(d*x + c)^4 + 1680*d*x*cos(d*x + c)^3 + 2520*d*x*cos(d* x + c)^2 + 1680*d*x*cos(d*x + c) + 420*d*x - (105*cos(d*x + c)^4 + 1184*co s(d*x + c)^3 + 2636*cos(d*x + c)^2 + 2236*cos(d*x + c) + 664)*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)
Time = 5.60 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.87 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} - \frac {3360 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {15 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {132 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {658 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {4340 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {6825 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Piecewise((-3360*d*x*tan(c/2 + d*x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*d*x/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 15 *tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 132*t an(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 658*tan (c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 4340*tan( c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 6825*tan(c /2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d), Ne(d, 0)), (x*c os(c)**5/(a*cos(c) + a)**4, True))
Time = 0.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{840 \, d} \]
1/840*(1680*sin(d*x + c)/((a^4 + a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)* (cos(d*x + c) + 1)) + (5145*sin(d*x + c)/(cos(d*x + c) + 1) - 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15 *sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 6720*arctan(sin(d*x + c)/(cos( d*x + c) + 1))/a^4)/d
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {3360 \, {\left (d x + c\right )}}{a^{4}} - \frac {1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
-1/840*(3360*(d*x + c)/a^4 - 1680*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2 *c)^2 + 1)*a^4) + (15*a^24*tan(1/2*d*x + 1/2*c)^7 - 147*a^24*tan(1/2*d*x + 1/2*c)^5 + 805*a^24*tan(1/2*d*x + 1/2*c)^3 - 5145*a^24*tan(1/2*d*x + 1/2* c))/a^28)/d
Time = 14.97 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-192\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+1144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-6112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )}{840\,a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]