Integrand size = 21, antiderivative size = 139 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {2 \sin (c+d x)}{9 a d (a+a \cos (c+d x))^4}+\frac {\sin (c+d x)}{15 a^2 d (a+a \cos (c+d x))^3}+\frac {2 \sin (c+d x)}{45 a^3 d (a+a \cos (c+d x))^2}+\frac {2 \sin (c+d x)}{45 d \left (a^5+a^5 \cos (c+d x)\right )} \]
1/9*sin(d*x+c)/d/(a+a*cos(d*x+c))^5-2/9*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4+ 1/15*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^3+2/45*sin(d*x+c)/a^3/d/(a+a*cos(d* x+c))^2+2/45*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Time = 2.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.47 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\left (2+10 \cos (c+d x)+21 \cos ^2(c+d x)+10 \cos ^3(c+d x)+2 \cos ^4(c+d x)\right ) \sin (c+d x)}{45 a^5 d (1+\cos (c+d x))^5} \]
((2 + 10*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])/(45*a^5*d*(1 + Cos[c + d*x])^5)
Time = 0.67 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 3237, 25, 3042, 3229, 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.
\(\displaystyle \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
\(\Big \downarrow \) 3237 |
\(\displaystyle \frac {\int -\frac {5 a-9 a \cos (c+d x)}{(\cos (c+d x) a+a)^4}dx}{9 a^2}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\int \frac {5 a-9 a \cos (c+d x)}{(\cos (c+d x) a+a)^4}dx}{9 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\int \frac {5 a-9 a \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}\) |
\(\Big \downarrow \) 3229 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \int \frac {1}{(\cos (c+d x) a+a)^3}dx}{9 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{9 a^2}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \left (\frac {2 \int \frac {1}{(\cos (c+d x) a+a)^2}dx}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{9 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \left (\frac {2 \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{9 a^2}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \left (\frac {2 \left (\frac {\int \frac {1}{\cos (c+d x) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{9 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \left (\frac {2 \left (\frac {\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}+\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}\right )}{9 a^2}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {\frac {2 a \sin (c+d x)}{d (a \cos (c+d x)+a)^4}-3 \left (\frac {\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {2 \left (\frac {\sin (c+d x)}{3 a d (a \cos (c+d x)+a)}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{5 a}\right )}{9 a^2}\) |
Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - ((2*a*Sin[c + d*x])/(d*(a + a* Cos[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)))/(9*a^2)
3.1.87.3.1 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[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*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)*(a*m - b*(2* m + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.82 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.32
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(45\) |
default | \(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\) | \(45\) |
parallelrisch | \(\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{720 a^{5} d}\) | \(47\) |
risch | \(\frac {4 i \left (30 \,{\mathrm e}^{6 i \left (d x +c \right )}+45 \,{\mathrm e}^{5 i \left (d x +c \right )}+81 \,{\mathrm e}^{4 i \left (d x +c \right )}+54 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{45 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(91\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d a}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{20 d a}-\frac {13 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{720 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{72 d a}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{4}}\) | \(152\) |
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 10 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{45 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
1/45*(2*cos(d*x + c)^4 + 10*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 = 3.42 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.49 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} - \frac {\tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{2}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]
Piecewise((tan(c/2 + d*x/2)**9/(144*a**5*d) - tan(c/2 + d*x/2)**5/(40*a**5 *d) + tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x*cos(c)**2/(a*cos(c) + a) **5, True))
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.48 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {18 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{720 \, a^{5} d} \]
1/720*(45*sin(d*x + c)/(cos(d*x + c) + 1) - 18*sin(d*x + c)^5/(cos(d*x + c ) + 1)^5 + 5*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(a^5*d)
Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.33 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{720 \, a^{5} d} \]
1/720*(5*tan(1/2*d*x + 1/2*c)^9 - 18*tan(1/2*d*x + 1/2*c)^5 + 45*tan(1/2*d *x + 1/2*c))/(a^5*d)
Time = 14.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.32 \[ \int \frac {\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+45\right )}{720\,a^5\,d} \]