\(\int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx\) [298]

Optimal result

Integrand size = 26, antiderivative size = 327 \[ \int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=\frac {x}{4 \sqrt [3]{2} \sqrt [3]{a}}-\frac {i \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} d}-\frac {i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {i \log (\cos (c+d x))}{4 \sqrt [3]{2} \sqrt [3]{a} d}+\frac {i \log (\tan (c+d x))}{6 \sqrt [3]{a} d}-\frac {i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a} d}-\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{4 \sqrt [3]{2} \sqrt [3]{a} d}-\frac {5 i}{2 d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}} \] Output:

1/8*x*2^(2/3)/a^(1/3)-1/3*I*arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^(1/3) 
)*3^(1/2)/a^(1/3))*3^(1/2)/a^(1/3)/d-1/4*I*3^(1/2)*arctan(1/3*(a^(1/3)+2^( 
2/3)*(a+I*a*tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))*2^(2/3)/a^(1/3)/d-1/8*I*ln 
(cos(d*x+c))*2^(2/3)/a^(1/3)/d+1/6*I*ln(tan(d*x+c))/a^(1/3)/d-1/2*I*ln(a^( 
1/3)-(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)/d-3/8*I*ln(2^(1/3)*a^(1/3)-(a+I*a*t 
an(d*x+c))^(1/3))*2^(2/3)/a^(1/3)/d-5/2*I/d/(a+I*a*tan(d*x+c))^(1/3)-cot(d 
*x+c)/d/(a+I*a*tan(d*x+c))^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=-\frac {\frac {8 i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}+\frac {6 i 2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {4 i \log (\tan (c+d x))}{\sqrt [3]{a}}-\frac {3 i 2^{2/3} \log (i+\tan (c+d x))}{\sqrt [3]{a}}+\frac {12 i \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{\sqrt [3]{a}}+\frac {9 i 2^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{\sqrt [3]{a}}+\frac {60 i}{\sqrt [3]{a+i a \tan (c+d x)}}+\frac {24 \cot (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}}}{24 d} \] Input:

Integrate[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(1/3),x]
 

Output:

-1/24*(((8*I)*Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(S 
qrt[3]*a^(1/3))])/a^(1/3) + ((6*I)*2^(2/3)*Sqrt[3]*ArcTan[(1 + (2^(2/3)*(a 
 + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - ((4*I)*Log[Tan[c 
+ d*x]])/a^(1/3) - ((3*I)*2^(2/3)*Log[I + Tan[c + d*x]])/a^(1/3) + ((12*I) 
*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/a^(1/3) + ((9*I)*2^(2/3)*Log 
[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/a^(1/3) + (60*I)/(a + I* 
a*Tan[c + d*x])^(1/3) + (24*Cot[c + d*x])/(a + I*a*Tan[c + d*x])^(1/3))/d
 

Rubi [A] (warning: unable to verify)

Time = 1.22 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.88, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {3042, 4044, 27, 3042, 4079, 27, 3042, 4083, 3042, 3962, 67, 16, 1082, 217, 4082, 67, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4044

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4079

\(\displaystyle -\frac {\frac {3 \int \frac {1}{3} \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (5 \tan (c+d x) a^2+2 i a^2\right )dx}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \cot (c+d x) (i \tan (c+d x) a+a)^{2/3} \left (5 \tan (c+d x) a^2+2 i a^2\right )dx}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {(i \tan (c+d x) a+a)^{2/3} \left (5 \tan (c+d x) a^2+2 i a^2\right )}{\tan (c+d x)}dx}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 4083

\(\displaystyle -\frac {\frac {3 a^2 \int (i \tan (c+d x) a+a)^{2/3}dx+2 i a \int \cot (c+d x) (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}dx}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 a^2 \int (i \tan (c+d x) a+a)^{2/3}dx+2 i a \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 3962

\(\displaystyle -\frac {\frac {2 i a \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^3 \int \frac {1}{(a-i a \tan (c+d x)) \sqrt [3]{i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 67

\(\displaystyle -\frac {\frac {2 i a \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^3 \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}+\frac {3 \int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {\frac {2 i a \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^3 \left (-\frac {3}{2} \int \frac {1}{-a^2 \tan ^2(c+d x)+i \sqrt [3]{2} a^{4/3} \tan (c+d x)+2^{2/3} a^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {2 i a \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^3 \left (\frac {3 \int \frac {1}{a^2 \tan ^2(c+d x)-3}d\left (i 2^{2/3} a^{2/3} \tan (c+d x)+1\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {2 i a \int \frac {(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{2/3}}{\tan (c+d x)}dx-\frac {3 i a^3 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 4082

\(\displaystyle -\frac {\frac {\frac {2 i a^3 \int \frac {\cot (c+d x)}{\sqrt [3]{i \tan (c+d x) a+a}}d\tan (c+d x)}{d}-\frac {3 i a^3 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 67

\(\displaystyle -\frac {\frac {\frac {2 i a^3 \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{i \tan (c+d x) a+a}}d\sqrt [3]{i \tan (c+d x) a+a}}{2 \sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^3 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {\frac {\frac {2 i a^3 \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{i \tan (c+d x) a+a} \sqrt [3]{a}+(i \tan (c+d x) a+a)^{2/3}}d\sqrt [3]{i \tan (c+d x) a+a}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^3 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {\frac {\frac {2 i a^3 \left (-\frac {3 \int \frac {1}{-(i \tan (c+d x) a+a)^{2/3}-3}d\left (\frac {2 \sqrt [3]{i \tan (c+d x) a+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^3 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\frac {\frac {2 i a^3 \left (\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log (\tan (c+d x))}{2 \sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{a}}\right )}{d}-\frac {3 i a^3 \left (-\frac {i \sqrt {3} \text {arctanh}\left (\frac {a \tan (c+d x)}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-i a \tan (c+d x)\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}+\frac {\log (a-i a \tan (c+d x))}{2 \sqrt [3]{2} \sqrt [3]{a}}\right )}{d}}{2 a^2}+\frac {15 i a}{2 d \sqrt [3]{a+i a \tan (c+d x)}}}{3 a}-\frac {\cot (c+d x)}{d \sqrt [3]{a+i a \tan (c+d x)}}\)

Input:

Int[Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(1/3),x]
 

Output:

-(Cot[c + d*x]/(d*(a + I*a*Tan[c + d*x])^(1/3))) - ((((-3*I)*a^3*(((-I)*Sq 
rt[3]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(1/3)*a^(1/3)) - (3*Log[2^(1/3 
)*a^(1/3) - I*a*Tan[c + d*x]])/(2*2^(1/3)*a^(1/3)) + Log[a - I*a*Tan[c + d 
*x]]/(2*2^(1/3)*a^(1/3))))/d + ((2*I)*a^3*((Sqrt[3]*ArcTan[(1 + (2*(a + I* 
a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[Tan[c + d*x]]/(2*a 
^(1/3)) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(1/3))))/d) 
/(2*a^2) + (((15*I)/2)*a)/(d*(a + I*a*Tan[c + d*x])^(1/3)))/(3*a)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 3042

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

rule 3962

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-b/d   S 
ubst[Int[(a + x)^(n - 1)/(a - x), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b 
, c, d, n}, 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 4079

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

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 [F]

\[\int \frac {\cot \left (d x +c \right )^{2}}{\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]

Input:

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

Output:

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

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. 717 vs. \(2 (231) = 462\).

Time = 0.10 (sec) , antiderivative size = 717, normalized size of antiderivative = 2.19 \[ \int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(2^(2/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*(-7*I*e^(4*I*d*x + 4*I*c) 
 - 4*I*e^(2*I*d*x + 2*I*c) + 3*I)*e^(4/3*I*d*x + 4/3*I*c) + 4*(a*d*e^(4*I* 
d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c))*(1/16*I/(a*d^3))^(1/3)*log(8*a*d^2 
*(1/16*I/(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2 
/3*I*d*x + 2/3*I*c)) + 4*(a*d*e^(4*I*d*x + 4*I*c) - a*d*e^(2*I*d*x + 2*I*c 
))*(1/27*I/(a*d^3))^(1/3)*log(9*a*d^2*(1/27*I/(a*d^3))^(2/3) + 2^(1/3)*(a/ 
(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 2*((-I*sqrt(3) 
*a*d + a*d)*e^(4*I*d*x + 4*I*c) + (I*sqrt(3)*a*d - a*d)*e^(2*I*d*x + 2*I*c 
))*(1/16*I/(a*d^3))^(1/3)*log(-4*(I*sqrt(3)*a*d^2 + a*d^2)*(1/16*I/(a*d^3) 
)^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I 
*c)) - 2*((I*sqrt(3)*a*d + a*d)*e^(4*I*d*x + 4*I*c) + (-I*sqrt(3)*a*d - a* 
d)*e^(2*I*d*x + 2*I*c))*(1/16*I/(a*d^3))^(1/3)*log(-4*(-I*sqrt(3)*a*d^2 + 
a*d^2)*(1/16*I/(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3 
)*e^(2/3*I*d*x + 2/3*I*c)) - 2*((-I*sqrt(3)*a*d + a*d)*e^(4*I*d*x + 4*I*c) 
 + (I*sqrt(3)*a*d - a*d)*e^(2*I*d*x + 2*I*c))*(1/27*I/(a*d^3))^(1/3)*log(- 
9/2*(I*sqrt(3)*a*d^2 + a*d^2)*(1/27*I/(a*d^3))^(2/3) + 2^(1/3)*(a/(e^(2*I* 
d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c)) - 2*((I*sqrt(3)*a*d + a* 
d)*e^(4*I*d*x + 4*I*c) + (-I*sqrt(3)*a*d - a*d)*e^(2*I*d*x + 2*I*c))*(1/27 
*I/(a*d^3))^(1/3)*log(-9/2*(-I*sqrt(3)*a*d^2 + a*d^2)*(1/27*I/(a*d^3))^(2/ 
3) + 2^(1/3)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=-\frac {i \, a {\left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {3 \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {6 \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} + \frac {8 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {12 \, {\left (-5 i \, a \tan \left (d x + c\right ) - 2 \, a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{2}} - \frac {4 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {8 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}}\right )}}{24 \, d} \] Input:

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

Output:

-1/24*I*a*(6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 
 2*(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(4/3) - 3*2^(2/3)*log(2^(2/3)* 
a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) 
 + a)^(2/3))/a^(4/3) + 6*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) 
+ a)^(1/3))/a^(4/3) + 8*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + 
a)^(1/3) + a^(1/3))/a^(1/3))/a^(4/3) - 12*(-5*I*a*tan(d*x + c) - 2*a)/((I* 
a*tan(d*x + c) + a)^(4/3)*a - (I*a*tan(d*x + c) + a)^(1/3)*a^2) - 4*log((I 
*a*tan(d*x + c) + a)^(2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3 
))/a^(4/3) + 8*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(4/3))/d
 

Giac [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=-\frac {\frac {6 i \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {3 i \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {6 i \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} + \frac {8 i \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {4 i \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {8 i \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {12 \, {\left (5 i \, a \tan \left (d x + c\right ) + 2 \, a\right )}}{i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} - i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a}}{24 \, d} \] Input:

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

Output:

-1/24*(6*I*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2 
*(I*a*tan(d*x + c) + a)^(1/3))/a^(1/3))/a^(1/3) - 3*I*2^(2/3)*log(2^(2/3)* 
a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + (I*a*tan(d*x + c) 
 + a)^(2/3))/a^(1/3) + 6*I*2^(2/3)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c 
) + a)^(1/3))/a^(1/3) + 8*I*sqrt(3)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c 
) + a)^(1/3) + a^(1/3))/a^(1/3))/a^(1/3) - 4*I*log((I*a*tan(d*x + c) + a)^ 
(2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/a^(1/3) + 8*I*log( 
(I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(1/3) - 12*(5*I*a*tan(d*x + c) + 
 2*a)/(I*(I*a*tan(d*x + c) + a)^(4/3) - I*(I*a*tan(d*x + c) + a)^(1/3)*a)) 
/d
 

Mupad [B] (verification not implemented)

Time = 1.76 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.71 \[ \int \frac {\cot ^2(c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

log(a^5*d*(a + a*tan(c + d*x)*1i)^(1/3)*36i - ((5832*a^7*d^6*(1i/(16*a*d^3 
))^(2/3) + 675*a^6*d^4*(a + a*tan(c + d*x)*1i)^(1/3))*(1i/(16*a*d^3))^(1/3 
) - a^6*d^3*315i)*(1i/(16*a*d^3))^(2/3))*(1i/(16*a*d^3))^(1/3) + log(a^5*d 
*(a + a*tan(c + d*x)*1i)^(1/3)*36i - ((5832*a^7*d^6*(1i/(27*a*d^3))^(2/3) 
+ 675*a^6*d^4*(a + a*tan(c + d*x)*1i)^(1/3))*(1i/(27*a*d^3))^(1/3) - a^6*d 
^3*315i)*(1i/(27*a*d^3))^(2/3))*(1i/(27*a*d^3))^(1/3) + (((a + a*tan(c + d 
*x)*1i)*5i)/(2*d) - (a*3i)/(2*d))/(a*(a + a*tan(c + d*x)*1i)^(1/3) - (a + 
a*tan(c + d*x)*1i)^(4/3)) + (log(((3^(1/2)*1i - 1)^2*(a^6*d^3*315i - ((3^( 
1/2)*1i - 1)*(675*a^6*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 1458*a^7*d^6*(3^ 
(1/2)*1i - 1)^2*(1i/(16*a*d^3))^(2/3))*(1i/(16*a*d^3))^(1/3))/2)*(1i/(16*a 
*d^3))^(2/3))/4 + a^5*d*(a + a*tan(c + d*x)*1i)^(1/3)*36i)*(3^(1/2)*1i - 1 
)*(1i/(16*a*d^3))^(1/3))/2 - (log(((3^(1/2)*1i + 1)^2*(a^6*d^3*315i + ((3^ 
(1/2)*1i + 1)*(675*a^6*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 1458*a^7*d^6*(3 
^(1/2)*1i + 1)^2*(1i/(16*a*d^3))^(2/3))*(1i/(16*a*d^3))^(1/3))/2)*(1i/(16* 
a*d^3))^(2/3))/4 + a^5*d*(a + a*tan(c + d*x)*1i)^(1/3)*36i)*(3^(1/2)*1i + 
1)*(1i/(16*a*d^3))^(1/3))/2 + (log(((3^(1/2)*1i - 1)^2*(a^6*d^3*315i - ((3 
^(1/2)*1i - 1)*(675*a^6*d^4*(a + a*tan(c + d*x)*1i)^(1/3) + 1458*a^7*d^6*( 
3^(1/2)*1i - 1)^2*(1i/(27*a*d^3))^(2/3))*(1i/(27*a*d^3))^(1/3))/2)*(1i/(27 
*a*d^3))^(2/3))/4 + a^5*d*(a + a*tan(c + d*x)*1i)^(1/3)*36i)*(3^(1/2)*1i - 
 1)*(1i/(27*a*d^3))^(1/3))/2 - (log(((3^(1/2)*1i + 1)^2*(a^6*d^3*315i +...
 

Reduce [F]

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

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

Output:

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