Integrand size = 24, antiderivative size = 133 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {25 x}{8 a^3}-\frac {25 \cot (c+d x)}{8 a^3 d}-\frac {3 i \log (\sin (c+d x))}{a^3 d}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 \cot (c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {3 \cot (c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )} \] Output:
-25/8*x/a^3-25/8*cot(d*x+c)/a^3/d-3*I*ln(sin(d*x+c))/a^3/d+1/6*cot(d*x+c)/ d/(a+I*a*tan(d*x+c))^3+11/24*cot(d*x+c)/a/d/(a+I*a*tan(d*x+c))^2+3/2*cot(d *x+c)/d/(a^3+I*a^3*tan(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.11 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {4 a^4 \cot (c+d x)+(a+i a \tan (c+d x)) \left (11 a^3 \cot (c+d x)+3 a (a+i a \tan (c+d x)) \left (-25 a (i+\cot (c+d x)) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )+12 a (\cot (c+d x)+2 (\log (\cos (c+d x))+\log (\tan (c+d x))) (-i+\tan (c+d x)))\right )\right )}{24 a^4 d (a+i a \tan (c+d x))^3} \] Input:
Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]
Output:
(4*a^4*Cot[c + d*x] + (a + I*a*Tan[c + d*x])*(11*a^3*Cot[c + d*x] + 3*a*(a + I*a*Tan[c + d*x])*(-25*a*(I + Cot[c + d*x])*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 12*a*(Cot[c + d*x] + 2*(Log[Cos[c + d*x]] + Log[Ta n[c + d*x]])*(-I + Tan[c + d*x])))))/(24*a^4*d*(a + I*a*Tan[c + d*x])^3)
Time = 1.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4042, 3042, 4079, 27, 3042, 4079, 27, 3042, 4012, 25, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^2 (a+i a \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4042 |
| \(\displaystyle \frac {\int \frac {\cot ^2(c+d x) (7 a-4 i a \tan (c+d x))}{(i \tan (c+d x) a+a)^2}dx}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {7 a-4 i a \tan (c+d x)}{\tan (c+d x)^2 (i \tan (c+d x) a+a)^2}dx}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {\int \frac {3 \cot ^2(c+d x) \left (13 a^2-11 i a^2 \tan (c+d x)\right )}{i \tan (c+d x) a+a}dx}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {\cot ^2(c+d x) \left (13 a^2-11 i a^2 \tan (c+d x)\right )}{i \tan (c+d x) a+a}dx}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \int \frac {13 a^2-11 i a^2 \tan (c+d x)}{\tan (c+d x)^2 (i \tan (c+d x) a+a)}dx}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int 2 \cot ^2(c+d x) \left (25 a^3-24 i a^3 \tan (c+d x)\right )dx}{2 a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \cot ^2(c+d x) \left (25 a^3-24 i a^3 \tan (c+d x)\right )dx}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {25 a^3-24 i a^3 \tan (c+d x)}{\tan (c+d x)^2}dx}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-\frac {25 a^3 \cot (c+d x)}{d}+\int -\cot (c+d x) \left (25 \tan (c+d x) a^3+24 i a^3\right )dx}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-\frac {25 a^3 \cot (c+d x)}{d}-\int \cot (c+d x) \left (25 \tan (c+d x) a^3+24 i a^3\right )dx}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-\frac {25 a^3 \cot (c+d x)}{d}-\int \frac {25 \tan (c+d x) a^3+24 i a^3}{\tan (c+d x)}dx}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-24 i a^3 \int \cot (c+d x)dx-\frac {25 a^3 \cot (c+d x)}{d}-25 a^3 x}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-24 i a^3 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {25 a^3 \cot (c+d x)}{d}-25 a^3 x}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (\frac {24 i a^3 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {25 a^3 \cot (c+d x)}{d}-25 a^3 x}{a^2}+\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {3 \left (\frac {12 a^2 \cot (c+d x)}{d (a+i a \tan (c+d x))}+\frac {-\frac {25 a^3 \cot (c+d x)}{d}-\frac {24 i a^3 \log (-\sin (c+d x))}{d}-25 a^3 x}{a^2}\right )}{4 a^2}+\frac {11 a \cot (c+d x)}{4 d (a+i a \tan (c+d x))^2}}{6 a^2}+\frac {\cot (c+d x)}{6 d (a+i a \tan (c+d x))^3}\) |
Input:
Int[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^3,x]
Output:
Cot[c + d*x]/(6*d*(a + I*a*Tan[c + d*x])^3) + ((11*a*Cot[c + d*x])/(4*d*(a + I*a*Tan[c + d*x])^2) + (3*((-25*a^3*x - (25*a^3*Cot[c + d*x])/d - ((24* I)*a^3*Log[-Sin[c + d*x]])/d)/a^2 + (12*a^2*Cot[c + d*x])/(d*(a + I*a*Tan[ c + d*x]))))/(4*a^2))/(6*a^2)
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_)])^(n_), x_Symbol] :> Simp[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) In t[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(a*A + b*B)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*( b*c - a*d))), x] + Simp[1/(2*a*m*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Free Q[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n, 0]
Time = 1.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86
| method | result | size |
| risch | \(-\frac {49 x}{8 a^{3}}-\frac {23 i {\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{3} d}-\frac {7 i {\mathrm e}^{-4 i \left (d x +c \right )}}{32 a^{3} d}-\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{48 a^{3} d}-\frac {6 c}{a^{3} d}-\frac {2 i}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {3 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(114\) |
| derivativedivides | \(-\frac {\cot \left (d x +c \right )}{a^{3} d}+\frac {9 i}{8 d \,a^{3} \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {3 i \ln \left (\cot \left (d x +c \right )^{2}+1\right )}{2 d \,a^{3}}+\frac {\frac {25 \pi }{16}-\frac {25 \,\operatorname {arccot}\left (\cot \left (d x +c \right )\right )}{8}}{a^{3} d}+\frac {1}{6 d \,a^{3} \left (i+\cot \left (d x +c \right )\right )^{3}}-\frac {31}{8 d \,a^{3} \left (i+\cot \left (d x +c \right )\right )}\) | \(115\) |
| default | \(-\frac {\cot \left (d x +c \right )}{a^{3} d}+\frac {9 i}{8 d \,a^{3} \left (i+\cot \left (d x +c \right )\right )^{2}}+\frac {3 i \ln \left (\cot \left (d x +c \right )^{2}+1\right )}{2 d \,a^{3}}+\frac {\frac {25 \pi }{16}-\frac {25 \,\operatorname {arccot}\left (\cot \left (d x +c \right )\right )}{8}}{a^{3} d}+\frac {1}{6 d \,a^{3} \left (i+\cot \left (d x +c \right )\right )^{3}}-\frac {31}{8 d \,a^{3} \left (i+\cot \left (d x +c \right )\right )}\) | \(115\) |
| norman | \(\frac {-\frac {1}{a d}-\frac {25 \tan \left (d x +c \right )^{4}}{3 a d}-\frac {25 \tan \left (d x +c \right )^{6}}{8 a d}-\frac {25 x \tan \left (d x +c \right )}{8 a}-\frac {75 x \tan \left (d x +c \right )^{3}}{8 a}-\frac {75 x \tan \left (d x +c \right )^{5}}{8 a}-\frac {25 x \tan \left (d x +c \right )^{7}}{8 a}-\frac {55 \tan \left (d x +c \right )^{2}}{8 a d}-\frac {3 i \tan \left (d x +c \right )^{5}}{2 a d}-\frac {35 i \tan \left (d x +c \right )}{12 d a}-\frac {15 i \tan \left (d x +c \right )^{3}}{4 d a}}{\tan \left (d x +c \right ) a^{2} \left (1+\tan \left (d x +c \right )^{2}\right )^{3}}-\frac {3 i \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {3 i \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \,a^{3}}\) | \(222\) |
Input:
int(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-49/8*x/a^3-23/16*I/a^3/d*exp(-2*I*(d*x+c))-7/32*I/a^3/d*exp(-4*I*(d*x+c)) -1/48*I/a^3/d*exp(-6*I*(d*x+c))-6/a^3/d*c-2*I/d/a^3/(exp(2*I*(d*x+c))-1)-3 *I/a^3/d*ln(exp(2*I*(d*x+c))-1)
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {588 \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 6 \, {\left (98 \, d x - 55 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 288 \, {\left (i \, e^{\left (8 i \, d x + 8 i \, c\right )} - i \, e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 117 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i}{96 \, {\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \] Input:
integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-1/96*(588*d*x*e^(8*I*d*x + 8*I*c) - 6*(98*d*x - 55*I)*e^(6*I*d*x + 6*I*c) + 288*(I*e^(8*I*d*x + 8*I*c) - I*e^(6*I*d*x + 6*I*c))*log(e^(2*I*d*x + 2* I*c) - 1) - 117*I*e^(4*I*d*x + 4*I*c) - 19*I*e^(2*I*d*x + 2*I*c) - 2*I)/(a ^3*d*e^(8*I*d*x + 8*I*c) - a^3*d*e^(6*I*d*x + 6*I*c))
Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\begin {cases} \frac {\left (- 35328 i a^{6} d^{2} e^{10 i c} e^{- 2 i d x} - 5376 i a^{6} d^{2} e^{8 i c} e^{- 4 i d x} - 512 i a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (\frac {\left (- 49 e^{6 i c} - 23 e^{4 i c} - 7 e^{2 i c} - 1\right ) e^{- 6 i c}}{8 a^{3}} + \frac {49}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {2 i}{a^{3} d e^{2 i c} e^{2 i d x} - a^{3} d} - \frac {49 x}{8 a^{3}} - \frac {3 i \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} \] Input:
integrate(cot(d*x+c)**2/(a+I*a*tan(d*x+c))**3,x)
Output:
Piecewise(((-35328*I*a**6*d**2*exp(10*I*c)*exp(-2*I*d*x) - 5376*I*a**6*d** 2*exp(8*I*c)*exp(-4*I*d*x) - 512*I*a**6*d**2*exp(6*I*c)*exp(-6*I*d*x))*exp (-12*I*c)/(24576*a**9*d**3), Ne(a**9*d**3*exp(12*I*c), 0)), (x*((-49*exp(6 *I*c) - 23*exp(4*I*c) - 7*exp(2*I*c) - 1)*exp(-6*I*c)/(8*a**3) + 49/(8*a** 3)), True)) - 2*I/(a**3*d*exp(2*I*c)*exp(2*I*d*x) - a**3*d) - 49*x/(8*a**3 ) - 3*I*log(exp(2*I*d*x) - exp(-2*I*c))/(a**3*d)
Exception generated. \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=-\frac {i \, \log \left (\tan \left (d x + c\right ) + i\right )}{16 \, a^{3} d} + \frac {49 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{16 \, a^{3} d} - \frac {3 i \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3} d} - \frac {i \, {\left (-75 i \, \tan \left (d x + c\right )^{3} - 189 \, \tan \left (d x + c\right )^{2} + 142 i \, \tan \left (d x + c\right ) + 24\right )}}{24 \, a^{3} d {\left (\tan \left (d x + c\right ) - i\right )}^{3} \tan \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
Output:
-1/16*I*log(tan(d*x + c) + I)/(a^3*d) + 49/16*I*log(tan(d*x + c) - I)/(a^3 *d) - 3*I*log(abs(tan(d*x + c)))/(a^3*d) - 1/24*I*(-75*I*tan(d*x + c)^3 - 189*tan(d*x + c)^2 + 142*I*tan(d*x + c) + 24)/(a^3*d*(tan(d*x + c) - I)^3* tan(d*x + c))
Time = 1.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,49{}\mathrm {i}}{16\,a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,3{}\mathrm {i}}{a^3\,d}+\frac {\frac {71\,\mathrm {tan}\left (c+d\,x\right )}{12\,a^3}-\frac {25\,{\mathrm {tan}\left (c+d\,x\right )}^3}{8\,a^3}-\frac {1{}\mathrm {i}}{a^3}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,63{}\mathrm {i}}{8\,a^3}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4-{\mathrm {tan}\left (c+d\,x\right )}^3\,3{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \] Input:
int(cot(c + d*x)^2/(a + a*tan(c + d*x)*1i)^3,x)
Output:
(log(tan(c + d*x) - 1i)*49i)/(16*a^3*d) - (log(tan(c + d*x) + 1i)*1i)/(16* a^3*d) - (log(tan(c + d*x))*3i)/(a^3*d) + ((71*tan(c + d*x))/(12*a^3) - 1i /a^3 + (tan(c + d*x)^2*63i)/(8*a^3) - (25*tan(c + d*x)^3)/(8*a^3))/(d*(tan (c + d*x)*1i - 3*tan(c + d*x)^2 - tan(c + d*x)^3*3i + tan(c + d*x)^4))
\[ \int \frac {\cot ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^2/(a+I*a*tan(d*x+c))^3,x)
Output:
( - 121*cos(c + d*x) - 148*int(tan((c + d*x)/2)**3/(tan((c + d*x)/2)**6*i + 6*tan((c + d*x)/2)**5 - 15*tan((c + d*x)/2)**4*i - 20*tan((c + d*x)/2)** 3 + 15*tan((c + d*x)/2)**2*i + 6*tan((c + d*x)/2) - i),x)*sin(c + d*x)*d + 560*int(tan((c + d*x)/2)**2/(tan((c + d*x)/2)**6*i + 6*tan((c + d*x)/2)** 5 - 15*tan((c + d*x)/2)**4*i - 20*tan((c + d*x)/2)**3 + 15*tan((c + d*x)/2 )**2*i + 6*tan((c + d*x)/2) - i),x)*sin(c + d*x)*d*i + 920*int(tan((c + d* x)/2)/(tan((c + d*x)/2)**6*i + 6*tan((c + d*x)/2)**5 - 15*tan((c + d*x)/2) **4*i - 20*tan((c + d*x)/2)**3 + 15*tan((c + d*x)/2)**2*i + 6*tan((c + d*x )/2) - i),x)*sin(c + d*x)*d + 60*int(1/(tan((c + d*x)/2)**8*i + 6*tan((c + d*x)/2)**7 - 15*tan((c + d*x)/2)**6*i - 20*tan((c + d*x)/2)**5 + 15*tan(( c + d*x)/2)**4*i + 6*tan((c + d*x)/2)**3 - tan((c + d*x)/2)**2*i),x)*sin(c + d*x)*d*i - 340*int(1/(tan((c + d*x)/2)**7*i + 6*tan((c + d*x)/2)**6 - 1 5*tan((c + d*x)/2)**5*i - 20*tan((c + d*x)/2)**4 + 15*tan((c + d*x)/2)**3* i + 6*tan((c + d*x)/2)**2 - tan((c + d*x)/2)*i),x)*sin(c + d*x)*d - 780*in t(1/(tan((c + d*x)/2)**6*i + 6*tan((c + d*x)/2)**5 - 15*tan((c + d*x)/2)** 4*i - 20*tan((c + d*x)/2)**3 + 15*tan((c + d*x)/2)**2*i + 6*tan((c + d*x)/ 2) - i),x)*sin(c + d*x)*d*i + 23*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x) *i - 43*log(tan((c + d*x)/2))*sin(c + d*x)*i - 67*sin(c + d*x)*d*x - 120)/ (sin(c + d*x)*a**3*d)