Integrand size = 24, antiderivative size = 112 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-2 a^2 x-\frac {2 a^2 \cot (c+d x)}{d}+\frac {i a^2 \cot ^2(c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {i a^2 \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \] Output:
-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 = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.14 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2 i a^2 \csc ^2(c+d x)}{d}-\frac {i a^2 \csc ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {a^2 \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {2 i a^2 \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]
Output:
((2*I)*a^2*Csc[c + d*x]^2)/d - ((I/2)*a^2*Csc[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*Co t[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (( 2*I)*a^2*Log[Sin[c + d*x]])/d
Time = 0.87 (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 |
| \(\displaystyle -\frac {a^2 \cot ^5(c+d x)}{5 d}+2 \left (-\frac {i a^2 \cot ^4(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {i a^2 \cot ^2(c+d x)}{2 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {i a^2 \log (-\sin (c+d x))}{d}-a^2 x\right )\) |
Input:
Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^2,x]
Output:
-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)
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 = 1.38 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77
| method | result | size |
| parallelrisch | \(-\frac {a^{2} \left (i \cot \left (d x +c \right )^{4}+\frac {2 \cot \left (d x +c \right )^{5}}{5}-2 i \cot \left (d x +c \right )^{2}-\frac {4 \cot \left (d x +c \right )^{3}}{3}-4 i \ln \left (\tan \left (d x +c \right )\right )+2 i \ln \left (\sec \left (d x +c \right )^{2}\right )+4 d x +4 \cot \left (d x +c \right )\right )}{2 d}\) | \(86\) |
| 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 | \(\frac {-a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 i a^{2} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(106\) |
| default | \(\frac {-a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 i a^{2} \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(106\) |
| norman | \(\frac {\frac {i a^{2} \tan \left (d x +c \right )^{3}}{d}-\frac {a^{2}}{5 d}-2 a^{2} x \tan \left (d x +c \right )^{5}+\frac {2 a^{2} \tan \left (d x +c \right )^{2}}{3 d}-\frac {2 a^{2} \tan \left (d x +c \right )^{4}}{d}-\frac {i a^{2} \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {2 i a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a^{2} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(134\) |
Input:
int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-1/2*a^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))/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 = 0.08 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.96 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (135 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 300 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 370 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 200 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 43 i \, a^{2} + 15 \, {\left (-i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \] Input:
integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-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 = 0.28 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.95 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2 i a^{2} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 270 i a^{2} e^{8 i c} e^{8 i d x} + 600 i a^{2} e^{6 i c} e^{6 i d x} - 740 i a^{2} e^{4 i c} e^{4 i d x} + 400 i a^{2} e^{2 i c} e^{2 i d x} - 86 i a^{2}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \] Input:
integrate(cot(d*x+c)**6*(a+I*a*tan(d*x+c))**2,x)
Output:
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 = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.97 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {60 \, {\left (d x + c\right )} a^{2} + 30 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a^{2} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a^{2} \tan \left (d x + c\right )^{4} - 30 i \, a^{2} \tan \left (d x + c\right )^{3} - 20 \, a^{2} \tan \left (d x + c\right )^{2} + 15 i \, a^{2} \tan \left (d x + c\right ) + 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \] Input:
integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-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
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.92 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 i \, a^{2} \log \left (\tan \left (d x + c\right ) + i\right )}{d} + \frac {2 i \, a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {60 \, a^{2} \tan \left (d x + c\right )^{4} - 30 i \, a^{2} \tan \left (d x + c\right )^{3} - 20 \, a^{2} \tan \left (d x + c\right )^{2} + 15 i \, a^{2} \tan \left (d x + c\right ) + 6 \, a^{2}}{30 \, d \tan \left (d x + c\right )^{5}} \] Input:
integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
-2*I*a^2*log(tan(d*x + c) + I)/d + 2*I*a^2*log(abs(tan(d*x + c)))/d - 1/30 *(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)/(d*tan(d*x + c)^5)
Time = 1.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {4\,a^2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{2}+\frac {a^2}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \] Input:
int(cot(c + d*x)^6*(a + a*tan(c + d*x)*1i)^2,x)
Output:
- (4*a^2*atan(2*tan(c + d*x) + 1i))/d - ((a^2*tan(c + d*x)*1i)/2 + a^2/5 - (2*a^2*tan(c + d*x)^2)/3 - a^2*tan(c + d*x)^3*1i + 2*a^2*tan(c + d*x)^4)/ (d*tan(c + d*x)^5)
Time = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.30 \[ \int \cot ^6(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^{2} \left (-688 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-48 \cos \left (d x +c \right )-480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{5} i +480 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{5} i -480 \sin \left (d x +c \right )^{5} d x -195 \sin \left (d x +c \right )^{5} i +480 \sin \left (d x +c \right )^{3} i -120 \sin \left (d x +c \right ) i \right )}{240 \sin \left (d x +c \right )^{5} d} \] Input:
int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^2,x)
Output:
(a**2*( - 688*cos(c + d*x)*sin(c + d*x)**4 + 256*cos(c + d*x)*sin(c + d*x) **2 - 48*cos(c + d*x) - 480*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**5*i + 480*log(tan((c + d*x)/2))*sin(c + d*x)**5*i - 480*sin(c + d*x)**5*d*x - 195*sin(c + d*x)**5*i + 480*sin(c + d*x)**3*i - 120*sin(c + d*x)*i))/(240 *sin(c + d*x)**5*d)