\(\int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx\) [727]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.29 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx=-\frac {2 a^2 \sqrt {\cot (c+d x)} \left (2 \sqrt [4]{-1} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}}{d} \] Input:

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

Output:

(-2*a^2*Sqrt[Cot[c + d*x]]*(2*(-1)^(1/4)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d* 
x]]] + 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 = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 4156, 3042, 4025, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4025

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4016

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 4025

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((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*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta 
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, 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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.88

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 290, normalized size of antiderivative = 5.92 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\frac {\sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 8 \, {\left (-i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{4 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.67 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx=-\frac {{\left (\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} + 4 \, a^{2} \sqrt {\tan \left (d x + c\right )}}{2 \, d} \] Input:

integrate(cot(d*x+c)^(1/2)*(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 = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (-\left (i + 1\right ) \, \sqrt {2} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right ) + \sqrt {\tan \left (d x + c\right )}\right )} a^{2}}{d} \] Input:

integrate(cot(d*x+c)^(1/2)*(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))) + s 
qrt(tan(d*x + c)))*a^2/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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