Integrand size = 19, antiderivative size = 143 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {5 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\sin (c+d x)}{21 a^2 d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{63 a d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 \sin (c+d x)}{63 d \left (a^5+a^5 \cos (c+d x)\right )} \] Output:
-1/9*sin(d*x+c)/d/(a+a*cos(d*x+c))^5+5/63*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^ 4+1/21*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3+2/63*sin(d*x+c)/a/d/(a^2+a^2*co s(d*x+c))^2+2/63*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.46 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\left (5+25 \cos (c+d x)+21 \cos ^2(c+d x)+10 \cos ^3(c+d x)+2 \cos ^4(c+d x)\right ) \sin (c+d x)}{63 a^5 d (1+\cos (c+d x))^5} \] Input:
Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^5,x]
Output:
((5 + 25*Cos[c + d*x] + 21*Cos[c + d*x]^2 + 10*Cos[c + d*x]^3 + 2*Cos[c + d*x]^4)*Sin[c + d*x])/(63*a^5*d*(1 + Cos[c + d*x])^5)
Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 3229, 3042, 3129, 3042, 3129, 3042, 3129, 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.
| \(\displaystyle \int \frac {\cos (c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle \frac {5 \int \frac {1}{(\cos (c+d x) a+a)^4}dx}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {1}{(\cos (c+d x) a+a)^3}dx}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \int \frac {1}{(\cos (c+d x) a+a)^2}dx}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{\cos (c+d x) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {5 \left (\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {3 \left (\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {2 \left (\frac {\sin (c+d x)}{3 a d (a \cos (c+d x)+a)}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}\right )}{7 a}\right )}{9 a}-\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
Input:
Int[Cos[c + d*x]/(a + a*Cos[c + d*x])^5,x]
Output:
-1/9*Sin[c + d*x]/(d*(a + a*Cos[c + d*x])^5) + (5*(Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) + (3*(Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) + (2*(S in[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(3*a*d*(a + a*Cos[ c + d*x]))))/(5*a)))/(7*a)))/(9*a)
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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.88 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.40
| method | result | size |
| parallelrisch | \(-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{7}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9\right )}{144 a^{5} d}\) | \(57\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(58\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(58\) |
| risch | \(\frac {4 i \left (63 \,{\mathrm e}^{5 i \left (d x +c \right )}+63 \,{\mathrm e}^{4 i \left (d x +c \right )}+84 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{63 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(80\) |
| norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{24 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{56 a d}-\frac {25 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1008 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{144 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) a^{4}}\) | \(133\) |
Input:
int(cos(d*x+c)/(a+a*cos(d*x+c))^5,x,method=_RETURNVERBOSE)
Output:
-1/144*tan(1/2*d*x+1/2*c)*(tan(1/2*d*x+1/2*c)^8+18/7*tan(1/2*d*x+1/2*c)^6- 6*tan(1/2*d*x+1/2*c)^2-9)/a^5/d
Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.86 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 25 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \] Input:
integrate(cos(d*x+c)/(a+a*cos(d*x+c))^5,x, algorithm="fricas")
Output:
1/63*(2*cos(d*x + c)^4 + 10*cos(d*x + c)^3 + 21*cos(d*x + c)^2 + 25*cos(d* x + c) + 5)*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 = 2.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.59 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} - \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} + \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)/(a+a*cos(d*x+c))**5,x)
Output:
Piecewise((-tan(c/2 + d*x/2)**9/(144*a**5*d) - tan(c/2 + d*x/2)**7/(56*a** 5*d) + tan(c/2 + d*x/2)**3/(24*a**5*d) + tan(c/2 + d*x/2)/(16*a**5*d), Ne( d, 0)), (x*cos(c)/(a*cos(c) + a)**5, True))
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {42 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {18 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{1008 \, a^{5} d} \] Input:
integrate(cos(d*x+c)/(a+a*cos(d*x+c))^5,x, algorithm="maxima")
Output:
1/1008*(63*sin(d*x + c)/(cos(d*x + c) + 1) + 42*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 18*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*d)
Time = 0.37 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.41 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{1008 \, a^{5} d} \] Input:
integrate(cos(d*x+c)/(a+a*cos(d*x+c))^5,x, algorithm="giac")
Output:
-1/1008*(7*tan(1/2*d*x + 1/2*c)^9 + 18*tan(1/2*d*x + 1/2*c)^7 - 42*tan(1/2 *d*x + 1/2*c)^3 - 63*tan(1/2*d*x + 1/2*c))/(a^5*d)
Time = 41.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.41 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+63\right )}{1008\,a^5\,d} \] Input:
int(cos(c + d*x)/(a + a*cos(c + d*x))^5,x)
Output:
(tan(c/2 + (d*x)/2)*(42*tan(c/2 + (d*x)/2)^2 - 18*tan(c/2 + (d*x)/2)^6 - 7 *tan(c/2 + (d*x)/2)^8 + 63))/(1008*a^5*d)
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.41 \[ \int \frac {\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+42 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+63\right )}{1008 a^{5} d} \] Input:
int(cos(d*x+c)/(a+a*cos(d*x+c))^5,x)
Output:
(tan((c + d*x)/2)*( - 7*tan((c + d*x)/2)**8 - 18*tan((c + d*x)/2)**6 + 42* tan((c + d*x)/2)**2 + 63))/(1008*a**5*d)