Integrand size = 21, antiderivative size = 107 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {x}{a^2}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {2 \csc (c+d x)}{a^2 d}-\frac {4 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d} \]
-x/a^2-cot(d*x+c)/a^2/d+1/3*cot(d*x+c)^3/a^2/d-2/5*cot(d*x+c)^5/a^2/d+2*cs c(d*x+c)/a^2/d-4/3*csc(d*x+c)^3/a^2/d+2/5*csc(d*x+c)^5/a^2/d
Time = 1.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\sec ^2(c+d x) \left (-120 d x \cos ^4\left (\frac {1}{2} (c+d x)\right )-31 \cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+\cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (15 \cot \left (\frac {1}{2} (c+d x)\right ) \csc \left (\frac {c}{2}\right )+193 \sec \left (\frac {c}{2}\right )\right ) \sin \left (\frac {d x}{2}\right )-31 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{30 a^2 d (1+\sec (c+d x))^2} \]
(Sec[c + d*x]^2*(-120*d*x*Cos[(c + d*x)/2]^4 - 31*Cos[(c + d*x)/2]*Sec[c/2 ]*Sin[(d*x)/2] + Cos[(c + d*x)/2]^3*(15*Cot[(c + d*x)/2]*Csc[c/2] + 193*Se c[c/2])*Sin[(d*x)/2] - 31*Cos[(c + d*x)/2]^2*Tan[c/2] + 3*Tan[(c + d*x)/2] ))/(30*a^2*d*(1 + Sec[c + d*x])^2)
Time = 0.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 4376, 3042, 4374, 2009}
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.
\(\displaystyle \int \frac {\cot ^2(c+d x)}{(a \sec (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
\(\Big \downarrow \) 4376 |
\(\displaystyle \frac {\int \cot ^6(c+d x) (a-a \sec (c+d x))^2dx}{a^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\cot \left (c+d x+\frac {\pi }{2}\right )^6}dx}{a^4}\) |
\(\Big \downarrow \) 4374 |
\(\displaystyle \frac {\int \left (a^2 \cot ^6(c+d x)-2 a^2 \csc (c+d x) \cot ^5(c+d x)+a^2 \csc ^2(c+d x) \cot ^4(c+d x)\right )dx}{a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {4 a^2 \csc ^3(c+d x)}{3 d}+\frac {2 a^2 \csc (c+d x)}{d}-a^2 x}{a^4}\) |
(-(a^2*x) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot [c + d*x]^5)/(5*d) + (2*a^2*Csc[c + d*x])/d - (4*a^2*Csc[c + d*x]^3)/(3*d) + (2*a^2*Csc[c + d*x]^5)/(5*d))/a^4
3.1.83.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Time = 0.66 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}\) | \(72\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-16 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}\) | \(72\) |
risch | \(-\frac {x}{a^{2}}+\frac {4 i \left (15 \,{\mathrm e}^{5 i \left (d x +c \right )}+30 \,{\mathrm e}^{4 i \left (d x +c \right )}+10 \,{\mathrm e}^{3 i \left (d x +c \right )}-35 \,{\mathrm e}^{2 i \left (d x +c \right )}-37 \,{\mathrm e}^{i \left (d x +c \right )}-13\right )}{15 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}\) | \(100\) |
1/8/d/a^2*(1/5*tan(1/2*d*x+1/2*c)^5-5/3*tan(1/2*d*x+1/2*c)^3+11*tan(1/2*d* x+1/2*c)-1/tan(1/2*d*x+1/2*c)-16*arctan(tan(1/2*d*x+1/2*c)))
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {26 \, \cos \left (d x + c\right )^{3} + 22 \, \cos \left (d x + c\right )^{2} + 15 \, {\left (d x \cos \left (d x + c\right )^{2} + 2 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 17 \, \cos \left (d x + c\right ) - 16}{15 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )} \]
-1/15*(26*cos(d*x + c)^3 + 22*cos(d*x + c)^2 + 15*(d*x*cos(d*x + c)^2 + 2* d*x*cos(d*x + c) + d*x)*sin(d*x + c) - 17*cos(d*x + c) - 16)/((a^2*d*cos(d *x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c))
\[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {\frac {165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{2}} - \frac {240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {15 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{2} \sin \left (d x + c\right )}}{120 \, d} \]
1/120*((165*sin(d*x + c)/(cos(d*x + c) + 1) - 25*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^2 - 240*arctan(sin(d *x + c)/(cos(d*x + c) + 1))/a^2 - 15*(cos(d*x + c) + 1)/(a^2*sin(d*x + c)) )/d
Time = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.79 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {120 \, {\left (d x + c\right )}}{a^{2}} + \frac {15}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 25 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 165 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}}}{120 \, d} \]
-1/120*(120*(d*x + c)/a^2 + 15/(a^2*tan(1/2*d*x + 1/2*c)) - (3*a^8*tan(1/2 *d*x + 1/2*c)^5 - 25*a^8*tan(1/2*d*x + 1/2*c)^3 + 165*a^8*tan(1/2*d*x + 1/ 2*c))/a^10)/d
Time = 14.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {x}{a^2}-\frac {\frac {26\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}-\frac {28\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {17\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{60}-\frac {1}{40}}{a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]