Integrand size = 21, antiderivative size = 147 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\cos ^2(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {\sin (c+d x)}{7 a d (a+a \cos (c+d x))^4}-\frac {17 \sin (c+d x)}{63 a^2 d (a+a \cos (c+d x))^3}+\frac {5 \sin (c+d x)}{63 a^3 d (a+a \cos (c+d x))^2}+\frac {5 \sin (c+d x)}{63 d \left (a^5+a^5 \cos (c+d x)\right )} \]
-1/9*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*cos(d*x+c))^5+1/7*sin(d*x+c)/a/d/(a+a* cos(d*x+c))^4-17/63*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3+5/63*sin(d*x+c)/a^ 3/d/(a+a*cos(d*x+c))^2+5/63*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 2.69 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\left (2+10 \cos (c+d x)+21 \cos ^2(c+d x)+25 \cos ^3(c+d x)+5 \cos ^4(c+d x)\right ) \sin (c+d x)}{63 a^5 d (1+\cos (c+d x))^5} \]
((2 + 10*Cos[c + d*x] + 21*Cos[c + d*x]^2 + 25*Cos[c + d*x]^3 + 5*Cos[c + d*x]^4)*Sin[c + d*x])/(63*a^5*d*(1 + Cos[c + d*x])^5)
Time = 0.84 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3244, 3042, 3447, 3042, 3498, 25, 3042, 3229, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x) (2 a-7 a \cos (c+d x))}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 a-7 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle -\frac {\int \frac {2 a \cos (c+d x)-7 a \cos ^2(c+d x)}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {2 a \sin \left (c+d x+\frac {\pi }{2}\right )-7 a \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3498 |
| \(\displaystyle -\frac {-\frac {\int -\frac {36 a^2-49 a^2 \cos (c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {36 a^2-49 a^2 \cos (c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {36 a^2-49 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle -\frac {\frac {\frac {17 a^2 \sin (c+d x)}{d (a \cos (c+d x)+a)^3}-15 a \int \frac {1}{(\cos (c+d x) a+a)^2}dx}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {17 a^2 \sin (c+d x)}{d (a \cos (c+d x)+a)^3}-15 a \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle -\frac {\frac {\frac {17 a^2 \sin (c+d x)}{d (a \cos (c+d x)+a)^3}-15 a \left (\frac {\int \frac {1}{\cos (c+d x) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {17 a^2 \sin (c+d x)}{d (a \cos (c+d x)+a)^3}-15 a \left (\frac {\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle -\frac {\frac {\frac {17 a^2 \sin (c+d x)}{d (a \cos (c+d x)+a)^3}-15 a \left (\frac {\sin (c+d x)}{3 a d (a \cos (c+d x)+a)}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}-\frac {9 a \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
-1/9*(Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^5) - ((-9*a*Sin [c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((17*a^2*Sin[c + d*x])/(d*(a + a *Cos[c + d*x])^3) - 15*a*(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]))))/(7*a^2))/(9*a^2)
3.1.86.3.1 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]
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)]
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_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b - a* B + b*C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[1 /(a^2*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*A*(m + 1) + m*(b *B - a*C) + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
Time = 0.84 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.39
| method | result | size |
| parallelrisch | \(-\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {18 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{144 a^{5} d}\) | \(57\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(58\) |
| default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(58\) |
| risch | \(\frac {2 i \left (63 \,{\mathrm e}^{7 i \left (d x +c \right )}+147 \,{\mathrm e}^{6 i \left (d x +c \right )}+315 \,{\mathrm e}^{5 i \left (d x +c \right )}+315 \,{\mathrm e}^{4 i \left (d x +c \right )}+273 \,{\mathrm e}^{3 i \left (d x +c \right )}+117 \,{\mathrm e}^{2 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )}{63 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(102\) |
| norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}-\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{112 d a}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1008 d a}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{336 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{336 d a}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} a^{4}}\) | \(171\) |
-1/144*(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)*tan(1/2*d*x+1/2*c)/a^5/d
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {{\left (5 \, \cos \left (d x + c\right )^{4} + 25 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 10 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{63 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/63*(5*cos(d*x + c)^4 + 25*cos(d*x + c)^3 + 21*cos(d*x + c)^2 + 10*cos(d* x + c) + 2)*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 = 4.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} - \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{5} d} - \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((-tan(c/2 + d*x/2)**9/(144*a**5*d) + tan(c/2 + d*x/2)**7/(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)**3/(a*cos(c) + a)**5, True))
Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {63 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {42 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {18 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {7 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{1008 \, a^{5} d} \]
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 = 0.43 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.40 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{1008 \, a^{5} d} \]
-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 = 14.41 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.39 \[ \int \frac {\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+42\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-63\right )}{1008\,a^5\,d} \]