\(\int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx\) [201]

Optimal result

Integrand size = 36, antiderivative size = 342 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}+\frac {a^{2/3} (2 i A+3 B) \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}-\frac {\sqrt {3} a^{2/3} (i A+B) \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}+\frac {a^{2/3} (2 i A+3 B) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d} \] Output:

1/4*a^(2/3)*(A-I*B)*x*2^(2/3)+1/3*a^(2/3)*(2*I*A+3*B)*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)/d-1/2*3^(1/2)*a^(2/3) 
*(I*A+B)*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)/d-1/4*a^(2/3)*(I*A+B)*ln(cos(d*x+c))*2^(2/3)/d-1/6*a^(2/3)*( 
2*I*A+3*B)*ln(tan(d*x+c))/d+1/2*a^(2/3)*(2*I*A+3*B)*ln(a^(1/3)-(a+I*a*tan( 
d*x+c))^(1/3))/d-3/4*a^(2/3)*(I*A+B)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c)) 
^(1/3))*2^(2/3)/d-A*cot(d*x+c)*(a+I*a*tan(d*x+c))^(2/3)/d
 

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.73 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\frac {2 a^2 (2 i A+3 B) \left (2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )-\log (\tan (c+d x))+3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )\right )-3 i 2^{2/3} a^2 (A-i B) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\log (i+\tan (c+d x))+3 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )\right )-12 a^{4/3} A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{12 a^{4/3} d} \] Input:

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

Output:

(2*a^2*((2*I)*A + 3*B)*(2*Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x 
])^(1/3))/(Sqrt[3]*a^(1/3))] - Log[Tan[c + d*x]] + 3*Log[a^(1/3) - (a + I* 
a*Tan[c + d*x])^(1/3)]) - (3*I)*2^(2/3)*a^2*(A - I*B)*(2*Sqrt[3]*ArcTan[(1 
 + (2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/Sqrt[3]] - Log[I + Tan[ 
c + d*x]] + 3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)]) - 12*a^ 
(4/3)*A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(2/3))/(12*a^(4/3)*d)
 

Rubi [A] (warning: unable to verify)

Time = 1.03 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.78, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 4081, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4081

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4083

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3962

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

(((3*I)*a^2*(A - I*B)*(((-I)*Sqrt[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 + (a^2*((2*I)* 
A + 3*B)*((Sqrt[3]*ArcTan[(1 + (2*(a + I*a*Tan[c + d*x])^(1/3))/a^(1/3))/S 
qrt[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)/(3*a) - (A*Cot[c + d*x]*(a + I*a 
*Tan[c + d*x])^(2/3))/d
 

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 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 [A] (verified)

Time = 0.22 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {3 i a^{2} \left (-\frac {\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) \left (-i B +A \right )}{a}+\frac {\frac {i A \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{3 a \tan \left (d x +c \right )}+\left (\frac {2 A}{3}-i B \right ) \left (\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right )}{a}\right )}{d}\)	\(294\)
default	\(\frac {3 i a^{2} \left (-\frac {\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) \left (-i B +A \right )}{a}+\frac {\frac {i A \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}}{3 a \tan \left (d x +c \right )}+\left (\frac {2 A}{3}-i B \right ) \left (\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}\right )}{a}\right )}{d}\)	\(294\)

Input:

int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(2/3)*(A+B*tan(d*x+c)),x,method=_RETUR 
NVERBOSE)
 

Output:

3*I/d*a^2*(-(1/6*2^(2/3)/a^(1/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/ 
3))-1/12*2^(2/3)/a^(1/3)*ln((a+I*a*tan(d*x+c))^(2/3)+2^(1/3)*a^(1/3)*(a+I* 
a*tan(d*x+c))^(1/3)+2^(2/3)*a^(2/3))+1/6*3^(1/2)*2^(2/3)/a^(1/3)*arctan(1/ 
3*3^(1/2)*(2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))*(A-I*B)/a+1/a*(1/ 
3*I*A*(a+I*a*tan(d*x+c))^(2/3)/a/tan(d*x+c)+(2/3*A-I*B)*(1/3/a^(1/3)*ln((a 
+I*a*tan(d*x+c))^(1/3)-a^(1/3))-1/6/a^(1/3)*ln((a+I*a*tan(d*x+c))^(2/3)+a^ 
(1/3)*(a+I*a*tan(d*x+c))^(1/3)+a^(2/3))+1/3*3^(1/2)/a^(1/3)*arctan(1/3*3^( 
1/2)*(2/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1)))))
 

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. 1096 vs. \(2 (255) = 510\).

Time = 0.15 (sec) , antiderivative size = 1096, normalized size of antiderivative = 3.20 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(6*2^(2/3)*(I*A*e^(2*I*d*x + 2*I*c) + I*A)*(a/(e^(2*I*d*x + 2*I*c) + 
1))^(2/3)*e^(4/3*I*d*x + 4/3*I*c) - 6*(1/2)^(1/3)*(d*e^(2*I*d*x + 2*I*c) - 
 d)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(1/3)*log((2^(1/3)*(A^2 
- 2*I*A*B - B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3* 
I*c) + 2*(1/2)^(2/3)*d^2*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(2/ 
3))/((A^2 - 2*I*A*B - B^2)*a)) - 3*(1/2)^(1/3)*((I*sqrt(3)*d - d)*e^(2*I*d 
*x + 2*I*c) - I*sqrt(3)*d + d)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^ 
3)^(1/3)*log((2^(1/3)*(A^2 - 2*I*A*B - B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1) 
)^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (1/2)^(2/3)*(I*sqrt(3)*d^2 + d^2)*((I*A^ 
3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(2/3))/((A^2 - 2*I*A*B - B^2)*a)) 
- 3*(1/2)^(1/3)*((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d) 
*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(1/3)*log((2^(1/3)*(A^2 - 2 
*I*A*B - B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c 
) - (1/2)^(2/3)*(-I*sqrt(3)*d^2 + d^2)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3 
)*a^2/d^3)^(2/3))/((A^2 - 2*I*A*B - B^2)*a)) - ((-I*sqrt(3)*d - d)*e^(2*I* 
d*x + 2*I*c) + I*sqrt(3)*d + d)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^ 
3)*a^2/d^3)^(1/3)*log(1/2*(2*2^(1/3)*(4*A^2 - 12*I*A*B - 9*B^2)*a*(a/(e^(2 
*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + (I*sqrt(3)*d^2 - d^2 
)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3)*a^2/d^3)^(2/3))/((4*A^2 - 1 
2*I*A*B - 9*B^2)*a)) - ((I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) - I*sqrt(...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.87 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=-\frac {i \, {\left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (A - i \, B\right )} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {3 \cdot 2^{\frac {2}{3}} {\left (A - i \, B\right )} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {6 \cdot 2^{\frac {2}{3}} {\left (A - i \, B\right )} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {4 \, \sqrt {3} {\left (2 \, A - 3 i \, B\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, {\left (2 \, A - 3 i \, B\right )} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} - \frac {4 \, {\left (2 \, A - 3 i \, B\right )} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {12 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} A}{a \tan \left (d x + c\right )}\right )} a}{12 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.37 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 5.36 (sec) , antiderivative size = 5825, normalized size of antiderivative = 17.03 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:

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

Output:

log((18*d^3*(A^3*a^9*19i - A*B^2*a^9*27i + 45*A^2*B*a^9) - (1458*a^7*d^6*( 
-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486* 
A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2 
*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 
- 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 
 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) - 9*d*(a + 
a*tan(c + d*x)*1i)^(1/3)*(135*B^2*a^8*d^3 - 75*A^2*a^8*d^3 + A*B*a^8*d^3*1 
98i))*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 
 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15 
066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5* 
B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12* 
1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3))*(-( 
(d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^ 
2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B 
^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 
16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1 
458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) + 9*d*(a + a* 
tan(c + d*x)*1i)^(1/3)*(A^5*a^10*16i + 27*B^5*a^10 + A*B^4*a^10*126i + 92* 
A^4*B*a^10 - 231*A^2*B^3*a^10 - A^3*B^2*a^10*208i))*(-((d^3*((((594*A^3*a^ 
12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2...
 

Reduce [F]

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

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

Output:

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