Integrand size = 12, antiderivative size = 143 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {4 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {4 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {8 \sin (c+d x)}{315 a d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {8 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
1/9*sin(d*x+c)/d/(a+a*cos(d*x+c))^5+4/63*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4 +4/105*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3+8/315*sin(d*x+c)/a/d/(a^2+a^2*c os(d*x+c))^2+8/315*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (126 \sin \left (\frac {1}{2} (c+d x)\right )+84 \sin \left (\frac {3}{2} (c+d x)\right )+36 \sin \left (\frac {5}{2} (c+d x)\right )+9 \sin \left (\frac {7}{2} (c+d x)\right )+\sin \left (\frac {9}{2} (c+d x)\right )\right )}{315 a^5 d (1+\cos (c+d x))^5} \]
(Cos[(c + d*x)/2]*(126*Sin[(c + d*x)/2] + 84*Sin[(3*(c + d*x))/2] + 36*Sin [(5*(c + d*x))/2] + 9*Sin[(7*(c + d*x))/2] + Sin[(9*(c + d*x))/2]))/(315*a ^5*d*(1 + Cos[c + d*x])^5)
Time = 0.58 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 3129, 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.
\(\displaystyle \int \frac {1}{(a \cos (c+d x)+a)^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {4 \int \frac {1}{(\cos (c+d x) a+a)^4}dx}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {4 \left (\frac {3 \int \frac {1}{(\cos (c+d x) a+a)^3}dx}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \left (\frac {3 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \int \frac {1}{(\cos (c+d x) a+a)^2}dx}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{\cos (c+d x) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{7 a}+\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\right )}{9 a}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}+\frac {4 \left (\frac {\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {3 \left (\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {2 \left (\frac {\sin (c+d x)}{3 a d (a \cos (c+d x)+a)}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}\right )}{7 a}\right )}{9 a}\) |
Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (4*(Sin[c + d*x]/(7*d*(a + a*C os[c + d*x])^4) + (3*(Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) + (2*(Sin[ 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)
3.1.89.3.1 Defintions of rubi rules used
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]
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.48
method | result | size |
risch | \(\frac {16 i \left (126 \,{\mathrm e}^{4 i \left (d x +c \right )}+84 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(69\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(71\) |
default | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(71\) |
parallelrisch | \(\frac {35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+180 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+378 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+420 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5040 a^{5} d}\) | \(73\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d a}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{28 d a}+\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d a}}{a^{4}}\) | \(99\) |
16/315*I*(126*exp(4*I*(d*x+c))+84*exp(3*I*(d*x+c))+36*exp(2*I*(d*x+c))+9*e xp(I*(d*x+c))+1)/d/a^5/(exp(I*(d*x+c))+1)^9
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {{\left (8 \, \cos \left (d x + c\right )^{4} + 40 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 100 \, \cos \left (d x + c\right ) + 83\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/315*(8*cos(d*x + c)^4 + 40*cos(d*x + c)^3 + 84*cos(d*x + c)^2 + 100*cos( d*x + c) + 83)*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 = 2.76 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{28 a^{5} d} + \frac {3 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{5} d} + \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{12 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((tan(c/2 + d*x/2)**9/(144*a**5*d) + tan(c/2 + d*x/2)**7/(28*a**5 *d) + 3*tan(c/2 + d*x/2)**5/(40*a**5*d) + tan(c/2 + d*x/2)**3/(12*a**5*d) + tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x/(a*cos(c) + a)**5, True))
Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \]
1/5040*(315*sin(d*x + c)/(cos(d*x + c) + 1) + 420*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 180*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*d)
Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.50 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{5040 \, a^{5} d} \]
1/5040*(35*tan(1/2*d*x + 1/2*c)^9 + 180*tan(1/2*d*x + 1/2*c)^7 + 378*tan(1 /2*d*x + 1/2*c)^5 + 420*tan(1/2*d*x + 1/2*c)^3 + 315*tan(1/2*d*x + 1/2*c)) /(a^5*d)
Time = 14.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+180\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]