\(\int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx\) [733]

Optimal result

Integrand size = 26, antiderivative size = 86 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {8 (-1)^{3/4} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 a^3}{3 d \sqrt {\cot (c+d x)}}-\frac {2 \left (i a^3+a^3 \cot (c+d x)\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \] Output:

8*(-1)^(3/4)*a^3*arctanh((-1)^(3/4)*cot(d*x+c)^(1/2))/d-16/3*a^3/d/cot(d*x 
+c)^(1/2)-2/3*(I*a^3+a^3*cot(d*x+c))/d/cot(d*x+c)^(3/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=-\frac {2 i a^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-12 (-1)^{3/4} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)} (-9 i+\tan (c+d x))\right )}{3 d} \] Input:

Integrate[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3,x]
 

Output:

(((-2*I)/3)*a^3*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-12*(-1)^(3/4)*ArcT 
an[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + Sqrt[Tan[c + d*x]]*(-9*I + Tan[c + d*x 
])))/d
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 4156, 3042, 4036, 27, 3042, 4074, 27, 3042, 4016, 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.

\(\displaystyle \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3dx\)

\(\Big \downarrow \) 4156

\(\displaystyle \int \frac {(a \cot (c+d x)+i a)^3}{\cot ^{\frac {5}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-a \tan \left (c+d x+\frac {\pi }{2}\right )+i a\right )^3}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4036

\(\displaystyle -\frac {2}{3} \int -\frac {2 (\cot (c+d x) a+i a) \left (\cot (c+d x) a^2+2 i a^2\right )}{\cot ^{\frac {3}{2}}(c+d x)}dx-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{3} \int \frac {(\cot (c+d x) a+i a) \left (\cot (c+d x) a^2+2 i a^2\right )}{\cot ^{\frac {3}{2}}(c+d x)}dx-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{3} \int \frac {\left (i a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (2 i a^2-a^2 \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 4074

\(\displaystyle \frac {4}{3} \left (-\frac {4 a^3}{d \sqrt {\cot (c+d x)}}+\int \frac {3 \left (\cot (c+d x) a^3+i a^3\right )}{\sqrt {\cot (c+d x)}}dx\right )-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{3} \left (-\frac {4 a^3}{d \sqrt {\cot (c+d x)}}+3 \int \frac {\cot (c+d x) a^3+i a^3}{\sqrt {\cot (c+d x)}}dx\right )-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{3} \left (-\frac {4 a^3}{d \sqrt {\cot (c+d x)}}+3 \int \frac {i a^3-a^3 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 4016

\(\displaystyle \frac {4}{3} \left (-\frac {4 a^3}{d \sqrt {\cot (c+d x)}}-\frac {6 a^6 \int \frac {1}{a^3 \cot (c+d x)-i a^3}d\sqrt {\cot (c+d x)}}{d}\right )-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {4}{3} \left (\frac {6 (-1)^{3/4} a^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {4 a^3}{d \sqrt {\cot (c+d x)}}\right )-\frac {2 \left (a^3 \cot (c+d x)+i a^3\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

Input:

Int[Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3,x]
 

Output:

(4*((6*(-1)^(3/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (4*a^3)/ 
(d*Sqrt[Cot[c + d*x]])))/3 - (2*(I*a^3 + a^3*Cot[c + d*x]))/(3*d*Cot[c + d 
*x]^(3/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4016

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2*(c^2/f)   Subst[Int[1/(b*c - d*x^2), x], x, Sqrt[b 
*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]
 

rule 4036

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])
 

rule 4074

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b 
*c - a*d)*(A*b - a*B)*((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*A*c 
+ b*B*c + A*b*d - a*B*d - (A*b*c - a*B*c - a*A*d - b*B*d)*Tan[e + f*x], x], 
 x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && LtQ[m 
, -1] && NeQ[a^2 + b^2, 0]
 

rule 4156

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Cot[e + f*x])^(m - n*p 
)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (71 ) = 142\).

Time = 0.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {2 i}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {6}{\sqrt {\cot \left (d x +c \right )}}-i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )-\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )\right )}{d}\)	\(201\)
default	\(\frac {a^{3} \left (-\frac {2 i}{3 \cot \left (d x +c \right )^{\frac {3}{2}}}-\frac {6}{\sqrt {\cot \left (d x +c \right )}}-i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )-\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )\right )}{d}\)	\(201\)

Input:

int(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*a^3*(-2/3*I/cot(d*x+c)^(3/2)-6/cot(d*x+c)^(1/2)-I*2^(1/2)*(ln((cot(d*x 
+c)+2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1))+2 
*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))) 
-2^(1/2)*(ln((cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)+2^(1/2)*c 
ot(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/ 
2)*cot(d*x+c)^(1/2))))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (70) = 140\).

Time = 0.08 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.97 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {3 \, \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 3 \, \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 16 \, {\left (-5 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

integrate(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
 

Output:

1/12*(3*sqrt(-64*I*a^6/d^2)*(d*e^(4*I*d*x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I* 
c) + d)*log(1/4*(8*I*a^3*e^(2*I*d*x + 2*I*c) + sqrt(-64*I*a^6/d^2)*(I*d*e^ 
(2*I*d*x + 2*I*c) - I*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2* 
I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/a^3) - 3*sqrt(-64*I*a^6/d^2)*(d*e^(4*I*d* 
x + 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)*log(1/4*(8*I*a^3*e^(2*I*d*x + 2* 
I*c) + sqrt(-64*I*a^6/d^2)*(-I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt((I*e^(2*I 
*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))*e^(-2*I*d*x - 2*I*c)/a^3) - 
 16*(-5*I*a^3*e^(4*I*d*x + 4*I*c) + I*a^3*e^(2*I*d*x + 2*I*c) + 4*I*a^3)*s 
qrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(d*e^(4*I*d*x 
+ 4*I*c) + 2*d*e^(2*I*d*x + 2*I*c) + d)
 

Sympy [F]

\[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=- i a^{3} \left (\int i \sqrt {\cot {\left (c + d x \right )}}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \sqrt {\cot {\left (c + d x \right )}}\right )\, dx\right ) \] Input:

integrate(cot(d*x+c)**(1/2)*(a+I*a*tan(d*x+c))**3,x)
 

Output:

-I*a**3*(Integral(I*sqrt(cot(c + d*x)), x) + Integral(-3*tan(c + d*x)*sqrt 
(cot(c + d*x)), x) + Integral(tan(c + d*x)**3*sqrt(cot(c + d*x)), x) + Int 
egral(-3*I*tan(c + d*x)**2*sqrt(cot(c + d*x)), x))
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).

Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=-\frac {3 \, {\left (\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - 2 \, {\left (-i \, a^{3} - \frac {9 \, a^{3}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{3 \, d} \] Input:

integrate(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")
 

Output:

-1/3*(3*((2*I + 2)*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + 
c)))) + (2*I + 2)*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + 
c)))) + (I - 1)*sqrt(2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 
1) - (I - 1)*sqrt(2)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) 
)*a^3 - 2*(-I*a^3 - 9*a^3/tan(d*x + c))*tan(d*x + c)^(3/2))/d
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.56 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=-\frac {2 \, {\left (-\left (6 i + 6\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + i \, \tan \left (d x + c\right )^{\frac {3}{2}} + 9 \, \sqrt {\tan \left (d x + c\right )}\right )} a^{3}}{3 \, d} \] Input:

integrate(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
 

Output:

-2/3*(-(6*I + 6)*sqrt(2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c))) 
 + I*tan(d*x + c)^(3/2) + 9*sqrt(tan(d*x + c)))*a^3/d
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\int \sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \] Input:

int(cot(c + d*x)^(1/2)*(a + a*tan(c + d*x)*1i)^3,x)
 

Output:

int(cot(c + d*x)^(1/2)*(a + a*tan(c + d*x)*1i)^3, x)
 

Reduce [F]

\[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^3 \, dx=a^{3} \left (\int \sqrt {\cot \left (d x +c \right )}d x -\left (\int \sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )^{3}d x \right ) i -3 \left (\int \sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}d x \right )+3 \left (\int \sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )d x \right ) i \right ) \] Input:

int(cot(d*x+c)^(1/2)*(a+I*a*tan(d*x+c))^3,x)
 

Output:

a**3*(int(sqrt(cot(c + d*x)),x) - int(sqrt(cot(c + d*x))*tan(c + d*x)**3,x 
)*i - 3*int(sqrt(cot(c + d*x))*tan(c + d*x)**2,x) + 3*int(sqrt(cot(c + d*x 
))*tan(c + d*x),x)*i)