Integrand size = 24, antiderivative size = 112 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-2 a^2 x-\frac {2 a^2 \cot (c+d x)}{d}+\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \]
-2*a^2*x-2*a^2*cot(d*x+c)/d+I*a^2*cot(d*x+c)^2/d+2/3*a^2*cot(d*x+c)^3/d-1/ 2*I*a^2*cot(d*x+c)^4/d-1/5*a^2*cot(d*x+c)^5/d+2*I*a^2*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.29 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {i a^2 \cot ^2(c+d x)}{d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {a^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {2 i a^2 \log (\cos (c+d x))}{d}+\frac {2 i a^2 \log (\tan (c+d x))}{d} \]
(I*a^2*Cot[c + d*x]^2)/d - ((I/2)*a^2*Cot[c + d*x]^4)/d - (a^2*Cot[c + d*x ]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) + (a^2*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + ((2*I) *a^2*Log[Cos[c + d*x]])/d + ((2*I)*a^2*Log[Tan[c + d*x]])/d
Time = 0.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 4025, 27, 3042, 4012, 25, 3042, 4012, 3042, 4012, 25, 3042, 4012, 3042, 4014, 3042, 25, 3956}
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.
\(\displaystyle \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^2}{\tan (c+d x)^6}dx\) |
\(\Big \downarrow \) 4025 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+\int 2 \cot ^5(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \int \cot ^5(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)^5}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (\int -\cot ^4(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\int \cot ^4(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\int \frac {i \tan (c+d x) a^2+a^2}{\tan (c+d x)^4}dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}\right )\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\int \cot ^3(c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}\right )\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\int -\cot ^2(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (\int \cot ^2(c+d x) \left (i \tan (c+d x) a^2+a^2\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (\int \frac {i \tan (c+d x) a^2+a^2}{\tan (c+d x)^2}dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}\right )\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (\int \cot (c+d x) \left (i a^2-a^2 \tan (c+d x)\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (\int \frac {i a^2-a^2 \tan (c+d x)}{\tan (c+d x)}dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}\right )\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (i a^2 \int \cot (c+d x)dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (i a^2 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-i a^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {i a^2 \log (-\sin (c+d x))}{d}-a^2 x\right )\) |
-1/5*(a^2*Cot[c + d*x]^5)/d + 2*(-(a^2*x) - (a^2*Cot[c + d*x])/d + ((I/2)* a^2*Cot[c + d*x]^2)/d + (a^2*Cot[c + d*x]^3)/(3*d) - ((I/4)*a^2*Cot[c + d* x]^4)/d + (I*a^2*Log[-Sin[c + d*x]])/d)
3.1.23.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.65 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77
method | result | size |
parallelrisch | \(-\frac {\left (i \left (\cot ^{4}\left (d x +c \right )\right )+\frac {2 \left (\cot ^{5}\left (d x +c \right )\right )}{5}-2 i \left (\cot ^{2}\left (d x +c \right )\right )-\frac {4 \left (\cot ^{3}\left (d x +c \right )\right )}{3}-4 i \ln \left (\tan \left (d x +c \right )\right )+2 i \ln \left (\sec ^{2}\left (d x +c \right )\right )+4 d x +4 \cot \left (d x +c \right )\right ) a^{2}}{2 d}\) | \(86\) |
risch | \(\frac {4 a^{2} c}{d}-\frac {2 i a^{2} \left (135 \,{\mathrm e}^{8 i \left (d x +c \right )}-300 \,{\mathrm e}^{6 i \left (d x +c \right )}+370 \,{\mathrm e}^{4 i \left (d x +c \right )}-200 \,{\mathrm e}^{2 i \left (d x +c \right )}+43\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(100\) |
derivativedivides | \(\frac {-a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 i a^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(106\) |
default | \(\frac {-a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 i a^{2} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(106\) |
norman | \(\frac {\frac {i a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{5 d}-2 a^{2} x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {2 a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {i a^{2} \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {2 i a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(134\) |
-1/2*(I*cot(d*x+c)^4+2/5*cot(d*x+c)^5-2*I*cot(d*x+c)^2-4/3*cot(d*x+c)^3-4* I*ln(tan(d*x+c))+2*I*ln(sec(d*x+c)^2)+4*d*x+4*cot(d*x+c))*a^2/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (100) = 200\).
Time = 0.23 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.96 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (135 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 300 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 370 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 200 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 43 i \, a^{2} + 15 \, {\left (-i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
-2/15*(135*I*a^2*e^(8*I*d*x + 8*I*c) - 300*I*a^2*e^(6*I*d*x + 6*I*c) + 370 *I*a^2*e^(4*I*d*x + 4*I*c) - 200*I*a^2*e^(2*I*d*x + 2*I*c) + 43*I*a^2 + 15 *(-I*a^2*e^(10*I*d*x + 10*I*c) + 5*I*a^2*e^(8*I*d*x + 8*I*c) - 10*I*a^2*e^ (6*I*d*x + 6*I*c) + 10*I*a^2*e^(4*I*d*x + 4*I*c) - 5*I*a^2*e^(2*I*d*x + 2* I*c) + I*a^2)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(10*I*d*x + 10*I*c) - 5*d *e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) - d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (100) = 200\).
Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.95 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2 i a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 270 i a^{2} e^{8 i c} e^{8 i d x} + 600 i a^{2} e^{6 i c} e^{6 i d x} - 740 i a^{2} e^{4 i c} e^{4 i d x} + 400 i a^{2} e^{2 i c} e^{2 i d x} - 86 i a^{2}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \]
2*I*a**2*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-270*I*a**2*exp(8*I*c)*exp(8 *I*d*x) + 600*I*a**2*exp(6*I*c)*exp(6*I*d*x) - 740*I*a**2*exp(4*I*c)*exp(4 *I*d*x) + 400*I*a**2*exp(2*I*c)*exp(2*I*d*x) - 86*I*a**2)/(15*d*exp(10*I*c )*exp(10*I*d*x) - 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I* d*x) - 150*d*exp(4*I*c)*exp(4*I*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d )
Time = 0.50 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.97 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {60 \, {\left (d x + c\right )} a^{2} + 30 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a^{2} \tan \left (d x + c\right )^{4} - 30 i \, a^{2} \tan \left (d x + c\right )^{3} - 20 \, a^{2} \tan \left (d x + c\right )^{2} + 15 i \, a^{2} \tan \left (d x + c\right ) + 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \]
-1/30*(60*(d*x + c)*a^2 + 30*I*a^2*log(tan(d*x + c)^2 + 1) - 60*I*a^2*log( tan(d*x + c)) + (60*a^2*tan(d*x + c)^4 - 30*I*a^2*tan(d*x + c)^3 - 20*a^2* tan(d*x + c)^2 + 15*I*a^2*tan(d*x + c) + 6*a^2)/tan(d*x + c)^5)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (100) = 200\).
Time = 1.03 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.89 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 55 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1920 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 960 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {-2192 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 55 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 i \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
1/480*(3*a^2*tan(1/2*d*x + 1/2*c)^5 - 15*I*a^2*tan(1/2*d*x + 1/2*c)^4 - 55 *a^2*tan(1/2*d*x + 1/2*c)^3 + 180*I*a^2*tan(1/2*d*x + 1/2*c)^2 - 1920*I*a^ 2*log(tan(1/2*d*x + 1/2*c) + I) + 960*I*a^2*log(tan(1/2*d*x + 1/2*c)) + 63 0*a^2*tan(1/2*d*x + 1/2*c) + (-2192*I*a^2*tan(1/2*d*x + 1/2*c)^5 - 630*a^2 *tan(1/2*d*x + 1/2*c)^4 + 180*I*a^2*tan(1/2*d*x + 1/2*c)^3 + 55*a^2*tan(1/ 2*d*x + 1/2*c)^2 - 15*I*a^2*tan(1/2*d*x + 1/2*c) - 3*a^2)/tan(1/2*d*x + 1/ 2*c)^5)/d
Time = 4.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {4\,a^2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{2}+\frac {a^2}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \]