\(\int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx\) [726]

Optimal result

Integrand size = 26, antiderivative size = 49 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {4 \sqrt [4]{-1} a^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2 \sqrt {\cot (c+d x)}}{d} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 a^2 \sqrt {\cot (c+d x)} \left (1+2 (-1)^{3/4} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}\right )}{d} \] Input:

Integrate[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2,x]
 

Output:

(-2*a^2*Sqrt[Cot[c + d*x]]*(1 + 2*(-1)^(3/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c 
+ d*x]]]*Sqrt[Tan[c + d*x]]))/d
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3042, 4156, 3042, 4026, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4016

\(\displaystyle -\frac {2 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {8 a^4 \int \frac {1}{2 i \cot (c+d x) a^2+2 a^2}d\sqrt {\cot (c+d x)}}{d}\)

\(\Big \downarrow \) 221

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

Input:

Int[Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2,x]
 

Output:

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

Defintions of rubi rules used

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 4026

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

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.88

method	result	size
derivativedivides	\(\frac {a^{2} \left (-2 \sqrt {\cot \left (d x +c \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}-\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}\right )}{d}\)	\(190\)
default	\(\frac {a^{2} \left (-2 \sqrt {\cot \left (d x +c \right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}-\frac {i \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{2}\right )}{d}\)	\(190\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 236, normalized size of antiderivative = 4.82 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {8 \, a^{2} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} - \sqrt {\frac {16 i \, a^{4}}{d^{2}}} d \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) + \sqrt {\frac {16 i \, a^{4}}{d^{2}}} d \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right )}{4 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

-a**2*(Integral(tan(c + d*x)**2*cot(c + d*x)**(3/2), x) + Integral(-2*I*ta 
n(c + d*x)*cot(c + d*x)**(3/2), x) + Integral(-cot(c + d*x)**(3/2), x))
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.67 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {{\left (-\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} - \frac {4 \, a^{2}}{\sqrt {\tan \left (d x + c\right )}}}{2 \, d} \] Input:

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

Output:

1/2*((-(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^2 - 4*a^2/sqrt(tan(d*x + c)))/d
                                                                                    
                                                                                    
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (-\left (i - 1\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \frac {1}{\sqrt {\tan \left (d x + c\right )}}\right )} a^{2}}{d} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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