\(\int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) [91]

Optimal result

Integrand size = 26, antiderivative size = 145 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {7 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{4 d}-\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {i \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d} \] Output:

7/4*a^(1/2)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d-2^(1/2)*a^(1/2)*ar 
ctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d-1/4*I*cot(d*x+c)*(a+ 
I*a*tan(d*x+c))^(1/2)/d-1/2*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.80 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {-7 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+4 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+\cot (c+d x) (i+2 \cot (c+d x)) \sqrt {a+i a \tan (c+d x)}}{4 d} \] Input:

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

Output:

-1/4*(-7*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]] + 4*Sqrt[2]*S 
qrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])] + Cot[c + d*x 
]*(I + 2*Cot[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/d
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 4044, 27, 3042, 4081, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4044

\(\displaystyle \frac {\int \frac {1}{2} \cot ^2(c+d x) (i a-3 a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4081

\(\displaystyle \frac {\frac {\int -\frac {1}{2} \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+7 a^2\right )dx}{a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \cot (c+d x) \sqrt {i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+7 a^2\right )dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {\sqrt {i \tan (c+d x) a+a} \left (i \tan (c+d x) a^2+7 a^2\right )}{\tan (c+d x)}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4083

\(\displaystyle \frac {-\frac {8 i a^2 \int \sqrt {i \tan (c+d x) a+a}dx+7 a \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {8 i a^2 \int \sqrt {i \tan (c+d x) a+a}dx+7 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3961

\(\displaystyle \frac {-\frac {\frac {16 a^3 \int \frac {1}{a-i a \tan (c+d x)}d\sqrt {i \tan (c+d x) a+a}}{d}+7 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {7 a \int \frac {(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)}dx+\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4082

\(\displaystyle \frac {-\frac {\frac {7 a^3 \int \frac {\cot (c+d x)}{\sqrt {i \tan (c+d x) a+a}}d\tan (c+d x)}{d}+\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {14 i a^2 \int \frac {1}{i-\frac {i (i \tan (c+d x) a+a)}{a}}d\sqrt {i \tan (c+d x) a+a}}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {\frac {8 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {14 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{d}}{2 a}-\frac {i a \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{d}}{4 a}-\frac {\cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{2 d}\)

Input:

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

Output:

-1/2*(Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/d + (-1/2*((-14*a^(5/2)*A 
rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (8*Sqrt[2]*a^(5/2)*ArcTanh 
[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d)/a - (I*a*Cot[c + d*x]*S 
qrt[a + I*a*Tan[c + d*x]])/d)/(4*a)
 

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 73

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]
 

rule 219

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])
 

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 3961

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]
 

rule 4044

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

rule 4081

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*d - B*c)*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(f*(n + 
1)*(c^2 + d^2))), x] - Simp[1/(a*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + 
f*x])^m*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c* 
m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 
0] && LtQ[n, -1]
 

rule 4082

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]
 

rule 4083

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]
 

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (115 ) = 230\).

Time = 11.06 (sec) , antiderivative size = 587, normalized size of antiderivative = 4.05

method	result	size
default	\(\frac {\sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (i \left (120 \cos \left (d x +c \right )^{2}+240 \cos \left (d x +c \right )+120\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}}\right )+i \left (105 \cos \left (d x +c \right )^{2}+210 \cos \left (d x +c \right )+105\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (105 \cos \left (d x +c \right )^{2}+210 \cos \left (d x +c \right )+105\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+\left (120 \cos \left (d x +c \right )^{2}+240 \cos \left (d x +c \right )+120\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}}{2}\right )+i \cot \left (d x +c \right ) \left (45 \cos \left (d x +c \right )^{2}+120 \cos \left (d x +c \right )+75\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\cot \left (d x +c \right ) \csc \left (d x +c \right ) \left (-34 \cos \left (d x +c \right )^{3}+50 \cos \left (d x +c \right )^{2}+154 \cos \left (d x +c \right )+70\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+i \cot \left (d x +c \right ) \left (30 \cos \left (d x +c \right )-30\right )+\cot \left (d x +c \right ) \csc \left (d x +c \right ) \left (-158 \cos \left (d x +c \right )^{3}+18 \cos \left (d x +c \right )^{2}+110 \cos \left (d x +c \right )+30\right )\right )}{120 d \left (\cos \left (d x +c \right )+1\right ) \left (1+\cos \left (d x +c \right )+i \sin \left (d x +c \right )\right )}\)	\(587\)

Input:

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

Output:

1/120/d*(a*(1+I*tan(d*x+c)))^(1/2)/(cos(d*x+c)+1)/(1+cos(d*x+c)+I*sin(d*x+ 
c))*(I*(120*cos(d*x+c)^2+240*cos(d*x+c)+120)*2^(1/2)*(-2*cos(d*x+c)/(cos(d 
*x+c)+1))^(1/2)*arctanh(2^(1/2)*(cot(d*x+c)-csc(d*x+c))/(cot(d*x+c)^2-2*co 
t(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))+I*(105*cos(d*x+c)^2+210*cos(d*x 
+c)+105)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((-2*cos(d*x+c)/(cos(d*x+c 
)+1))^(1/2)-cot(d*x+c)+csc(d*x+c))+(105*cos(d*x+c)^2+210*cos(d*x+c)+105)*( 
-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d 
*x+c)+1))^(1/2))+(120*cos(d*x+c)^2+240*cos(d*x+c)+120)*2^(1/2)*(-2*cos(d*x 
+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)* 
2^(1/2))+I*cot(d*x+c)*(45*cos(d*x+c)^2+120*cos(d*x+c)+75)*2^(1/2)*(-cos(d* 
x+c)/(cos(d*x+c)+1))^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+cot(d*x+c) 
*csc(d*x+c)*(-34*cos(d*x+c)^3+50*cos(d*x+c)^2+154*cos(d*x+c)+70)*2^(1/2)*( 
-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+I*c 
ot(d*x+c)*(30*cos(d*x+c)-30)+cot(d*x+c)*csc(d*x+c)*(-158*cos(d*x+c)^3+18*c 
os(d*x+c)^2+110*cos(d*x+c)+30))
 

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. 519 vs. \(2 (110) = 220\).

Time = 0.09 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.58 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/16*(8*sqrt(2)*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqr 
t(a/d^2)*log(4*((d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 
1))*sqrt(a/d^2) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) - 8*sqrt(2)*(d*e^(4 
*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/d^2)*log(-4*((d*e^(2 
*I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(a/d^2) - a*e^( 
I*d*x + I*c))*e^(-I*d*x - I*c)) - 7*(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d* 
x + 2*I*c) + d)*sqrt(a/d^2)*log(16*(3*a^2*e^(2*I*d*x + 2*I*c) + 2*sqrt(2)* 
(a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c 
) + 1))*sqrt(a/d^2) + a^2)*e^(-2*I*d*x - 2*I*c)) + 7*(d*e^(4*I*d*x + 4*I*c 
) - 2*d*e^(2*I*d*x + 2*I*c) + d)*sqrt(a/d^2)*log(16*(3*a^2*e^(2*I*d*x + 2* 
I*c) - 2*sqrt(2)*(a*d*e^(3*I*d*x + 3*I*c) + a*d*e^(I*d*x + I*c))*sqrt(a/(e 
^(2*I*d*x + 2*I*c) + 1))*sqrt(a/d^2) + a^2)*e^(-2*I*d*x - 2*I*c)) - 4*sqrt 
(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(3*e^(5*I*d*x + 5*I*c) + 4*e^(3*I*d* 
x + 3*I*c) + e^(I*d*x + I*c)))/(d*e^(4*I*d*x + 4*I*c) - 2*d*e^(2*I*d*x + 2 
*I*c) + d)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.23 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {a^{2} {\left (\frac {2 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} + \sqrt {i \, a \tan \left (d x + c\right ) + a} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} + a^{3}} + \frac {4 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} - \frac {7 \, \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )}}{8 \, d} \] Input:

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

Output:

1/8*a^2*(2*((I*a*tan(d*x + c) + a)^(3/2) + sqrt(I*a*tan(d*x + c) + a)*a)/( 
(I*a*tan(d*x + c) + a)^2*a - 2*(I*a*tan(d*x + c) + a)*a^2 + a^3) + 4*sqrt( 
2)*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)))/a^(3/2) - 7*log((sqrt(I*a*tan(d*x + c) + a) - 
 sqrt(a))/(sqrt(I*a*tan(d*x + c) + a) + sqrt(a)))/a^(3/2))/d
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.86 \[ \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{4\,d\,{\mathrm {tan}\left (c+d\,x\right )}^2}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,7{}\mathrm {i}}{4\,d}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}}{4\,a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^2}+\frac {\sqrt {2}\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,\sqrt {a}}\right )\,1{}\mathrm {i}}{d} \] Input:

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

Output:

(2^(1/2)*a^(1/2)*atan((2^(1/2)*(a + a*tan(c + d*x)*1i)^(1/2)*1i)/(2*a^(1/2 
)))*1i)/d - (a^(1/2)*atan(((a + a*tan(c + d*x)*1i)^(1/2)*1i)/a^(1/2))*7i)/ 
(4*d) - (a + a*tan(c + d*x)*1i)^(3/2)/(4*a*d*tan(c + d*x)^2) - (a + a*tan( 
c + d*x)*1i)^(1/2)/(4*d*tan(c + d*x)^2)
 

Reduce [F]

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

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

Output:

sqrt(a)*int(sqrt(tan(c + d*x)*i + 1)*cot(c + d*x)**3,x)