\(\int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx\) [720]

Optimal result

Integrand size = 24, antiderivative size = 45 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2 \sqrt [4]{-1} a \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a \sqrt {\cot (c+d x)}}{d} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {2 a \left (\sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )+\sqrt {\cot (c+d x)}\right )}{d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 4156, 3042, 4011, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4016

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(-1)^(1/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*a*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 4011

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int 
[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*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] && GtQ[m, 0]
 

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 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 = 0.23 (sec) , antiderivative size = 260, normalized size of antiderivative = 5.78

method	result	size
derivativedivides	\(\frac {a \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (i \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+2 i \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 i \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-2 \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-2 \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )-8\right )}{4 d}\)	\(260\)
default	\(\frac {a \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right ) \left (i \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right )+2 i \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 i \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-2 \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-2 \sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )-8\right )}{4 d}\)	\(260\)

Input:

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

Output:

1/4*a/d*(1/tan(d*x+c))^(3/2)*tan(d*x+c)*(I*2^(1/2)*tan(d*x+c)^(1/2)*ln(-(t 
an(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c) 
-1))+2*I*2^(1/2)*tan(d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*I*2 
^(1/2)*tan(d*x+c)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-2*2^(1/2)*tan( 
d*x+c)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))-2*2^(1/2)*tan(d*x+c)^(1/2) 
*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))-2^(1/2)*tan(d*x+c)^(1/2)*ln(-(2^(1/2) 
*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))-8 
)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 228, normalized size of antiderivative = 5.07 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {d \sqrt {\frac {4 i \, a^{2}}{d^{2}}} \log \left (\frac {{\left ({\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {4 i \, a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} + 2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a}\right ) - d \sqrt {\frac {4 i \, a^{2}}{d^{2}}} \log \left (-\frac {{\left ({\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {4 i \, a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} - 2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a}\right ) - 8 \, a \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{4 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.82 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx=\frac {{\left (-\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a - \frac {8 \, a}{\sqrt {\tan \left (d x + c\right )}}}{4 \, d} \] Input:

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

Output:

1/4*((-(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 - 8*a/sqrt(tan(d*x + c)))/d
                                                                                    
                                                                                    
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx=-\frac {i \, {\left (-\left (i + 1\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) - \frac {2 i}{\sqrt {\tan \left (d x + c\right )}}\right )} a}{d} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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