Integrand size = 21, antiderivative size = 119 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {3 x}{a^3}+\frac {9 \sin (c+d x)}{5 a^3 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {3 \cos ^2(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}+\frac {3 \sin (c+d x)}{d \left (a^3+a^3 \cos (c+d x)\right )} \] Output:
-3*x/a^3+9/5*sin(d*x+c)/a^3/d-1/5*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c ))^3-3/5*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^2+3*sin(d*x+c)/d/(a^ 3+a^3*cos(d*x+c))
Time = 0.84 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\sin (c+d x) \left (120 \arcsin (\cos (c+d x)) \cos ^6\left (\frac {1}{2} (c+d x)\right )+\left (24+57 \cos (c+d x)+39 \cos ^2(c+d x)+5 \cos ^3(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{5 a^3 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{7/2}} \] Input:
Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^3,x]
Output:
(Sin[c + d*x]*(120*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^6 + (24 + 57*Cos[ c + d*x] + 39*Cos[c + d*x]^2 + 5*Cos[c + d*x]^3)*Sqrt[Sin[c + d*x]^2]))/(5 *a^3*d*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^(7/2))
Time = 0.90 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3244, 27, 3042, 3456, 27, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {3 \cos ^2(c+d x) (a-2 a \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int \frac {\cos ^2(c+d x) (a-2 a \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {3 \cos (c+d x) \left (2 a^2-3 a^2 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {\cos (c+d x) \left (2 a^2-3 a^2 \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 a^2-3 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {2 a^2 \cos (c+d x)-3 a^2 \cos ^2(c+d x)}{\cos (c+d x) a+a}dx}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {\int \frac {2 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )-3 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {3 \left (\frac {\frac {\int \frac {5 a^3 \cos (c+d x)}{\cos (c+d x) a+a}dx}{a}-\frac {3 a \sin (c+d x)}{d}}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (\frac {5 a^2 \int \frac {\cos (c+d x)}{\cos (c+d x) a+a}dx-\frac {3 a \sin (c+d x)}{d}}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {5 a^2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-\frac {3 a \sin (c+d x)}{d}}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {3 \left (\frac {5 a^2 \left (\frac {x}{a}-\int \frac {1}{\cos (c+d x) a+a}dx\right )-\frac {3 a \sin (c+d x)}{d}}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (\frac {5 a^2 \left (\frac {x}{a}-\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx\right )-\frac {3 a \sin (c+d x)}{d}}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {3 \left (\frac {5 a^2 \left (\frac {x}{a}-\frac {\sin (c+d x)}{d (a \cos (c+d x)+a)}\right )-\frac {3 a \sin (c+d x)}{d}}{a^2}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)^2}\right )}{5 a^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
Input:
Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^3,x]
Output:
-1/5*(Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^3) - (3*((a*Cos [c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^2) + ((-3*a*Sin[c + d*x] )/d + 5*a^2*(x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))))/a^2))/(5*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[(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 = 0.92 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {\frac {243 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {26 \cos \left (2 d x +2 c \right )}{81}+\frac {5 \cos \left (3 d x +3 c \right )}{243}+\frac {58}{81}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{80}-3 d x}{a^{3} d}\) | \(66\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-24 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(85\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-24 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(85\) |
| risch | \(-\frac {3 x}{a^{3}}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}+\frac {4 i \left (15 \,{\mathrm e}^{4 i \left (d x +c \right )}+50 \,{\mathrm e}^{3 i \left (d x +c \right )}+70 \,{\mathrm e}^{2 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}+12\right )}{5 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(112\) |
| norman | \(\frac {-\frac {3 x}{a}+\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {45 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a d}+\frac {591 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {81 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 a d}+\frac {51 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{20 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{10 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{20 a d}-\frac {12 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {18 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {12 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} a^{2}}\) | \(226\) |
Input:
int(cos(d*x+c)^4/(a+a*cos(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
3/80*(81*tan(1/2*d*x+1/2*c)*(cos(d*x+c)+26/81*cos(2*d*x+2*c)+5/243*cos(3*d *x+3*c)+58/81)*sec(1/2*d*x+1/2*c)^4-80*d*x)/a^3/d
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {15 \, d x \cos \left (d x + c\right )^{3} + 45 \, d x \cos \left (d x + c\right )^{2} + 45 \, d x \cos \left (d x + c\right ) + 15 \, d x - {\left (5 \, \cos \left (d x + c\right )^{3} + 39 \, \cos \left (d x + c\right )^{2} + 57 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right )}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^3,x, algorithm="fricas")
Output:
-1/5*(15*d*x*cos(d*x + c)^3 + 45*d*x*cos(d*x + c)^2 + 45*d*x*cos(d*x + c) + 15*d*x - (5*cos(d*x + c)^3 + 39*cos(d*x + c)^2 + 57*cos(d*x + c) + 24)*s in(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos( d*x + c) + a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (109) = 218\).
Time = 2.09 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.02 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\begin {cases} - \frac {60 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} - \frac {60 d x}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} - \frac {9 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} + \frac {75 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} + \frac {125 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{20 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 20 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**4/(a+a*cos(d*x+c))**3,x)
Output:
Piecewise((-60*d*x*tan(c/2 + d*x/2)**2/(20*a**3*d*tan(c/2 + d*x/2)**2 + 20 *a**3*d) - 60*d*x/(20*a**3*d*tan(c/2 + d*x/2)**2 + 20*a**3*d) + tan(c/2 + d*x/2)**7/(20*a**3*d*tan(c/2 + d*x/2)**2 + 20*a**3*d) - 9*tan(c/2 + d*x/2) **5/(20*a**3*d*tan(c/2 + d*x/2)**2 + 20*a**3*d) + 75*tan(c/2 + d*x/2)**3/( 20*a**3*d*tan(c/2 + d*x/2)**2 + 20*a**3*d) + 125*tan(c/2 + d*x/2)/(20*a**3 *d*tan(c/2 + d*x/2)**2 + 20*a**3*d), Ne(d, 0)), (x*cos(c)**4/(a*cos(c) + a )**3, True))
Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{20 \, d} \] Input:
integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^3,x, algorithm="maxima")
Output:
1/20*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(co s(d*x + c) + 1)) + (85*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 120*arc tan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {\frac {60 \, {\left (d x + c\right )}}{a^{3}} - \frac {40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 85 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{20 \, d} \] Input:
integrate(cos(d*x+c)^4/(a+a*cos(d*x+c))^3,x, algorithm="giac")
Output:
-1/20*(60*(d*x + c)/a^3 - 40*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^3) - (a^12*tan(1/2*d*x + 1/2*c)^5 - 10*a^12*tan(1/2*d*x + 1/2*c)^3 + 85*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 41.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+96\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-60\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{20\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \] Input:
int(cos(c + d*x)^4/(a + a*cos(c + d*x))^3,x)
Output:
(sin(c/2 + (d*x)/2) - 12*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 96*cos( c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2) + 40*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d *x)/2) - 60*cos(c/2 + (d*x)/2)^5*(c + d*x))/(20*a^3*d*cos(c/2 + (d*x)/2)^5 )
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+75 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d x +125 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-60 d x}{20 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(cos(d*x+c)^4/(a+a*cos(d*x+c))^3,x)
Output:
(tan((c + d*x)/2)**7 - 9*tan((c + d*x)/2)**5 + 75*tan((c + d*x)/2)**3 - 60 *tan((c + d*x)/2)**2*d*x + 125*tan((c + d*x)/2) - 60*d*x)/(20*a**3*d*(tan( (c + d*x)/2)**2 + 1))