Integrand size = 21, antiderivative size = 124 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {5 x}{a^2}+\frac {12 \sin (c+d x)}{a^2 d}-\frac {5 \cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {10 \cos ^3(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {\cos ^4(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {4 \sin ^3(c+d x)}{a^2 d} \] Output:
-5*x/a^2+12*sin(d*x+c)/a^2/d-5*cos(d*x+c)*sin(d*x+c)/a^2/d-10/3*cos(d*x+c) ^3*sin(d*x+c)/a^2/d/(1+cos(d*x+c))-1/3*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos( d*x+c))^2-4*sin(d*x+c)^3/a^2/d
Time = 0.95 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\sin (c+d x) \left (60 \arcsin (\cos (c+d x)) \cos ^4\left (\frac {1}{2} (c+d x)\right )+\left (24+33 \cos (c+d x)+6 \cos ^2(c+d x)-\cos ^3(c+d x)+\cos ^4(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{3 a^2 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{5/2}} \] Input:
Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^2,x]
Output:
(Sin[c + d*x]*(60*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^4 + (24 + 33*Cos[c + d*x] + 6*Cos[c + d*x]^2 - Cos[c + d*x]^3 + Cos[c + d*x]^4)*Sqrt[Sin[c + d*x]^2]))/(3*a^2*d*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^(5/2))
Time = 0.70 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3244, 27, 3042, 3456, 27, 3042, 3227, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{(a \cos (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {2 \cos ^3(c+d x) (2 a-3 a \cos (c+d x))}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^3(c+d x) (2 a-3 a \cos (c+d x))}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (2 a-3 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {2 \left (\frac {\int 3 \cos ^2(c+d x) \left (5 a^2-6 a^2 \cos (c+d x)\right )dx}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {3 \int \cos ^2(c+d x) \left (5 a^2-6 a^2 \cos (c+d x)\right )dx}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {3 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (5 a^2-6 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {2 \left (\frac {3 \left (5 a^2 \int \cos ^2(c+d x)dx-6 a^2 \int \cos ^3(c+d x)dx\right )}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {3 \left (5 a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-6 a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {2 \left (\frac {3 \left (\frac {6 a^2 \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}+5 a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (\frac {3 \left (5 a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {6 a^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {2 \left (\frac {3 \left (5 a^2 \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {6 a^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {2 \left (\frac {3 \left (\frac {6 a^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}+5 a^2 \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{d (\cos (c+d x)+1)}\right )}{3 a^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}\) |
Input:
Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^2,x]
Output:
-1/3*(Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^2) - (2*((5*Cos [c + d*x]^3*Sin[c + d*x])/(d*(1 + Cos[c + d*x])) + (3*(5*a^2*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) + (6*a^2*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/ d))/a^2))/(3*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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
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 = 1.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62
| method | result | size |
| parallelrisch | \(\frac {43 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {14 \cos \left (2 d x +2 c \right )}{129}-\frac {\cos \left (3 d x +3 c \right )}{129}+\frac {\cos \left (4 d x +4 c \right )}{258}+\frac {73}{86}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-40 d x}{8 a^{2} d}\) | \(77\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-20 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(101\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8 \left (-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-20 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(101\) |
| risch | \(-\frac {5 x}{a^{2}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{2} d}-\frac {15 i {\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}+\frac {15 i {\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{2} d}+\frac {2 i \left (15 \,{\mathrm e}^{2 i \left (d x +c \right )}+27 \,{\mathrm e}^{i \left (d x +c \right )}+14\right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}+\frac {\sin \left (3 d x +3 c \right )}{12 a^{2} d}\) | \(143\) |
| norman | \(\frac {-\frac {5 x}{a}+\frac {21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {143 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {521 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 a d}+\frac {230 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}+\frac {185 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{6 a d}-\frac {25 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {50 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {50 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {25 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a}}{a \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(243\) |
Input:
int(cos(d*x+c)^5/(a+a*cos(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/8*(43*tan(1/2*d*x+1/2*c)*(cos(d*x+c)+14/129*cos(2*d*x+2*c)-1/129*cos(3*d *x+3*c)+1/258*cos(4*d*x+4*c)+73/86)*sec(1/2*d*x+1/2*c)^2-40*d*x)/a^2/d
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {15 \, d x \cos \left (d x + c\right )^{2} + 30 \, d x \cos \left (d x + c\right ) + 15 \, d x - {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 33 \, \cos \left (d x + c\right ) + 24\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \] Input:
integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^2,x, algorithm="fricas")
Output:
-1/3*(15*d*x*cos(d*x + c)^2 + 30*d*x*cos(d*x + c) + 15*d*x - (cos(d*x + c) ^4 - cos(d*x + c)^3 + 6*cos(d*x + c)^2 + 33*cos(d*x + c) + 24)*sin(d*x + c ))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (117) = 234\).
Time = 2.52 (sec) , antiderivative size = 700, normalized size of antiderivative = 5.65 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)**5/(a+a*cos(d*x+c))**2,x)
Output:
Piecewise((-30*d*x*tan(c/2 + d*x/2)**6/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18* a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 9 0*d*x*tan(c/2 + d*x/2)**4/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/ 2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 90*d*x*tan(c/2 + d*x/2)**2/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 30*d*x/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - tan(c/2 + d*x/2)**9/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2* d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 24*tan (c/2 + d*x/2)**7/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2 )**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 138*tan(c/2 + d*x/2)**5 /(6*a**2*d*tan(c/2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d *tan(c/2 + d*x/2)**2 + 6*a**2*d) + 160*tan(c/2 + d*x/2)**3/(6*a**2*d*tan(c /2 + d*x/2)**6 + 18*a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2 )**2 + 6*a**2*d) + 63*tan(c/2 + d*x/2)/(6*a**2*d*tan(c/2 + d*x/2)**6 + 18* a**2*d*tan(c/2 + d*x/2)**4 + 18*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d), Ne (d, 0)), (x*cos(c)**5/(a*cos(c) + a)**2, True))
Time = 0.18 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.67 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {60 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^2,x, algorithm="maxima")
Output:
1/6*(4*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^3/(cos(d*x + c ) + 1)^3 + 15*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^2 + 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a^ 2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (27*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 60*arctan(sin(d*x + c)/(co s(d*x + c) + 1))/a^2)/d
Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {\frac {30 \, {\left (d x + c\right )}}{a^{2}} - \frac {4 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \] Input:
integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^2,x, algorithm="giac")
Output:
-1/6*(30*(d*x + c)/a^2 - 4*(15*tan(1/2*d*x + 1/2*c)^5 + 20*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)^3*a^2) + ( a^4*tan(1/2*d*x + 1/2*c)^3 - 27*a^4*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 42.00 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=-\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-60\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+30\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (c+d\,x\right )}{6\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \] Input:
int(cos(c + d*x)^5/(a + a*cos(c + d*x))^2,x)
Output:
-(sin(c/2 + (d*x)/2) - 28*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) - 60*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) - 16*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 30*cos(c/2 + (d*x)/ 2)^3*(c + d*x))/(6*a^2*d*cos(c/2 + (d*x)/2)^3)
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx=\frac {-\cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+9 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d x +\cos \left (d x +c \right )+2 \sin \left (d x +c \right )^{4}+23 \sin \left (d x +c \right )^{2}-15 \sin \left (d x +c \right ) d x -1}{3 \sin \left (d x +c \right ) a^{2} d \left (\cos \left (d x +c \right )+1\right )} \] Input:
int(cos(d*x+c)^5/(a+a*cos(d*x+c))^2,x)
Output:
( - cos(c + d*x)*sin(c + d*x)**4 + 9*cos(c + d*x)*sin(c + d*x)**2 - 15*cos (c + d*x)*sin(c + d*x)*d*x + cos(c + d*x) + 2*sin(c + d*x)**4 + 23*sin(c + d*x)**2 - 15*sin(c + d*x)*d*x - 1)/(3*sin(c + d*x)*a**2*d*(cos(c + d*x) + 1))