Integrand size = 26, antiderivative size = 141 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {3 i a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {i a^2}{d \sqrt {a+i a \tan (c+d x)}}-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}} \]
-3*I*a^(3/2)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d+2*I*a^(3/2)*arcta nh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d-I*a^2/d/(a+I*a* tan(d*x+c))^(1/2)-a^2*cot(d*x+c)/d/(a+I*a*tan(d*x+c))^(1/2)
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {-3 i a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+2 i \sqrt {2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )-a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d} \]
((-3*I)*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + (2*I)*Sqrt[2 ]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] - a*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d
Time = 0.98 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4036, 27, 3042, 4079, 3042, 4083, 3042, 3961, 219, 4082, 73, 221}
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.
| \(\displaystyle \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (c+d x))^{3/2}}{\tan (c+d x)^2}dx\) |
| \(\Big \downarrow \) 4036 |
| \(\displaystyle -\int -\frac {\cot (c+d x) \left (3 i a^2-5 a^2 \tan (c+d x)\right )}{2 \sqrt {i \tan (c+d x) a+a}}dx-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {\cot (c+d x) \left (3 i a^2-5 a^2 \tan (c+d x)\right )}{\sqrt {i \tan (c+d x) a+a}}dx-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {3 i a^2-5 a^2 \tan (c+d x)}{\tan (c+d x) \sqrt {i \tan (c+d x) a+a}}dx-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4079 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (3 i a^3-a^3 \tan (c+d x)\right )dx}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (3 i a^3-a^3 \tan (c+d x)\right )}{\tan (c+d x)}dx}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4083 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 i a^2 \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx-4 a^3 \int \sqrt {i \tan (c+d x) a+a}dx}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 i a^2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx-4 a^3 \int \sqrt {i \tan (c+d x) a+a}dx}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3961 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {8 i a^4 \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}+3 i a^2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 i a^2 \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx+\frac {4 i \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {3 i a^4 \int \frac {\cot (c+d x)}{\sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}+\frac {4 i \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {6 a^3 \int \frac {1}{i-\frac {i (i \tan (c+d x) a+a)}{a}}d\sqrt {i \tan (c+d x) a+a}}{d}+\frac {4 i \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {4 i \sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {6 i a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}}{a^2}-\frac {2 i a^2}{d \sqrt {a+i a \tan (c+d x)}}\right )-\frac {a^2 \cot (c+d x)}{d \sqrt {a+i a \tan (c+d x)}}\) |
-((a^2*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]])) + ((((-6*I)*a^(7/2)*A rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + ((4*I)*Sqrt[2]*a^(7/2)*Arc Tanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d)/a^2 - ((2*I)*a^2)/( d*Sqrt[a + I*a*Tan[c + d*x]]))/2
3.1.97.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a , b, c, d}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x] )^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] + Si mp[a/(d*(b*c + a*d)*(n + 1)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[ e + f*x])^(n + 1)*Simp[b*(b*c*(m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[n, -1] && (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]
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[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[( A*b + a*B)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m, x], x] - Simp[(B*c - A *d)/(b*c + a*d) Int[(a + b*Tan[e + f*x])^m*((a - b*Tan[e + f*x])/(c + d*T an[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Time = 1.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {2 i a^{3} \left (-\frac {-\frac {i \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )}{d}\) | \(99\) |
| default | \(\frac {2 i a^{3} \left (-\frac {-\frac {i \sqrt {a +i a \tan \left (d x +c \right )}}{2 a \tan \left (d x +c \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )}{d}\) | \(99\) |
2*I/d*a^3*(-1/a*(-1/2*I*(a+I*a*tan(d*x+c))^(1/2)/a/tan(d*x+c)+3/2/a^(1/2)* arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2)))+1/a^(3/2)*2^(1/2)*arctanh(1/2*( a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (108) = 216\).
Time = 0.25 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.55 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {4 \, \sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \log \left (\frac {4 \, {\left (a^{2} e^{\left (i \, d x + i \, c\right )} + {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) - 4 \, \sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \log \left (\frac {4 \, {\left (a^{2} e^{\left (i \, d x + i \, c\right )} + {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a}\right ) + 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, \sqrt {2} {\left (i \, d e^{\left (3 i \, d x + 3 i \, c\right )} + i \, d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \log \left (16 \, {\left (3 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, \sqrt {2} {\left (-i \, d e^{\left (3 i \, d x + 3 i \, c\right )} - i \, d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {-\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + a^{2}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + 4 \, \sqrt {2} {\left (i \, a e^{\left (3 i \, d x + 3 i \, c\right )} + i \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{4 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
-1/4*(4*sqrt(2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(-a^3/d^2)*log(4*(a^2*e^(I *d*x + I*c) + (I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt(-a^3/d^2)*sqrt(a/(e^(2* I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/a) - 4*sqrt(2)*(d*e^(2*I*d*x + 2*I* c) - d)*sqrt(-a^3/d^2)*log(4*(a^2*e^(I*d*x + I*c) + (-I*d*e^(2*I*d*x + 2*I *c) - I*d)*sqrt(-a^3/d^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I *c)/a) + 3*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(-a^3/d^2)*log(16*(3*a^2*e^(2*I *d*x + 2*I*c) - 2*sqrt(2)*(I*d*e^(3*I*d*x + 3*I*c) + I*d*e^(I*d*x + I*c))* sqrt(-a^3/d^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)) + a^2)*e^(-2*I*d*x - 2*I* c)) - 3*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(-a^3/d^2)*log(16*(3*a^2*e^(2*I*d* x + 2*I*c) - 2*sqrt(2)*(-I*d*e^(3*I*d*x + 3*I*c) - I*d*e^(I*d*x + I*c))*sq rt(-a^3/d^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)) + a^2)*e^(-2*I*d*x - 2*I*c) ) + 4*sqrt(2)*(I*a*e^(3*I*d*x + 3*I*c) + I*a*e^(I*d*x + I*c))*sqrt(a/(e^(2 *I*d*x + 2*I*c) + 1)))/(d*e^(2*I*d*x + 2*I*c) - d)
\[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=-\frac {i \, {\left (2 \, \sqrt {2} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) - 3 \, \sqrt {a} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right ) - \frac {2 i \, \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\tan \left (d x + c\right )}\right )} a}{2 \, d} \]
-1/2*I*(2*sqrt(2)*sqrt(a)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a))) - 3*sqrt(a)*log((sqrt( I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a))) - 2*I*sqrt(I*a*tan(d*x + c) + a)/tan(d*x + c))*a/d
\[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{2} \,d x } \]
Time = 4.62 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{a^2}\right )\,\sqrt {-a^3}\,3{}\mathrm {i}}{d}-\frac {a\,\mathrm {cot}\left (c+d\,x\right )\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{d}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-a^3}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,a^2}\right )\,\sqrt {-a^3}\,2{}\mathrm {i}}{d} \]