\(\int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\) [284]

Optimal result

Integrand size = 24, antiderivative size = 254 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i a^{4/3} x}{2^{2/3}}-\frac {\sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}+\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {a^{4/3} \log (\tan (c+d x))}{2 d}+\frac {3 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d} \] Output:

-1/2*I*a^(4/3)*x*2^(1/3)-3^(1/2)*a^(4/3)*arctan(1/3*(a^(1/3)+2*(a+I*a*tan( 
d*x+c))^(1/3))*3^(1/2)/a^(1/3))/d+2^(1/3)*3^(1/2)*a^(4/3)*arctan(1/3*(a^(1 
/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))*3^(1/2)/a^(1/3))/d-1/2*a^(4/3)*ln(co 
s(d*x+c))*2^(1/3)/d-1/2*a^(4/3)*ln(tan(d*x+c))/d+3/2*a^(4/3)*ln(a^(1/3)-(a 
+I*a*tan(d*x+c))^(1/3))/d-3/2*a^(4/3)*ln(2^(1/3)*a^(1/3)-(a+I*a*tan(d*x+c) 
)^(1/3))*2^(1/3)/d
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.12 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {a^{4/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+2 \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-\sqrt [3]{2} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )\right )}{2 d} \] Input:

Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.11, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 4045, 3042, 3959, 3042, 3962, 69, 16, 1082, 217, 4077, 27, 3042, 4082, 69, 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.

\(\displaystyle \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+i a \tan (c+d x))^{4/3}}{\tan (c+d x)}dx\)

\(\Big \downarrow \) 4045

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

\(\Big \downarrow \) 3042

\(\displaystyle i \int (i \tan (c+d x) a+a)^{4/3}dx-\frac {i \int \frac {(i \tan (c+d x) a+a)^{4/3} (\tan (c+d x) a+i a)}{\tan (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3959

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3962

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 4077

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4082

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

Int[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3),x]
 

Output:

I*(((-2*I)*a^2*((I*Sqrt[3]*ArcTanh[(a*Tan[c + d*x])/Sqrt[3]])/(2^(2/3)*a^( 
2/3)) - (3*Log[2^(1/3)*a^(1/3) - I*a*Tan[c + d*x]])/(2*2^(2/3)*a^(2/3)) + 
Log[a - I*a*Tan[c + d*x]]/(2*2^(2/3)*a^(2/3))))/d + ((3*I)*a*(a + I*a*Tan[ 
c + d*x])^(1/3))/d) - (I*((I*a^3*(-((Sqrt[3]*ArcTan[(1 + (2*(a + I*a*Tan[c 
 + d*x])^(1/3))/a^(1/3))/Sqrt[3]])/a^(2/3)) - Log[Tan[c + d*x]]/(2*a^(2/3) 
) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(2/3))))/d + ((3* 
I)*a^2*(a + I*a*Tan[c + d*x])^(1/3))/d))/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 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   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 3959

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + 
b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[2*a   Int[(a + b*Tan[c + d* 
x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && GtQ[n 
, 1]
 

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 4045

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

rule 4077

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

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}+\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{d}-\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 d}-\frac {a^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}\)	\(245\)
default	\(-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}+\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{d}-\frac {a^{\frac {4}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 d}-\frac {a^{\frac {4}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}\)	\(245\)

Input:

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^(4/3),x,method=_RETURNVERBOSE)
 

Output:

-1/d*a^(4/3)*2^(1/3)*ln((a+I*a*tan(d*x+c))^(1/3)-2^(1/3)*a^(1/3))+1/2/d*a^ 
(4/3)*2^(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/d*a^(4/3)*2^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*( 
2^(2/3)/a^(1/3)*(a+I*a*tan(d*x+c))^(1/3)+1))+1/d*a^(4/3)*ln((a+I*a*tan(d*x 
+c))^(1/3)-a^(1/3))-1/2/d*a^(4/3)*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/d*a^(4/3)*3^(1/2)*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. 446 vs. \(2 (188) = 376\).

Time = 0.09 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.76 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="fricas")
 

Output:

1/2*2^(1/3)*(-a^4/d^3)^(1/3)*(I*sqrt(3) - 1)*log(1/2*(2*2^(1/3)*a*(a/(e^(2 
*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + 2^(1/3)*(I*sqrt(3)*d 
 - d)*(-a^4/d^3)^(1/3))/a) + 1/2*2^(1/3)*(-a^4/d^3)^(1/3)*(-I*sqrt(3) - 1) 
*log(1/2*(2*2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2 
/3*I*c) + 2^(1/3)*(-I*sqrt(3)*d - d)*(-a^4/d^3)^(1/3))/a) + 1/2*(a^4/d^3)^ 
(1/3)*(-I*sqrt(3) - 1)*log(1/2*(2*2^(1/3)*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 + d)*(a^4/d^3)^(1/3))/a) + 1/ 
2*(a^4/d^3)^(1/3)*(I*sqrt(3) - 1)*log(1/2*(2*2^(1/3)*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 + d)*(a^4/d^3)^(1 
/3))/a) + 2^(1/3)*(-a^4/d^3)^(1/3)*log((2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) 
+ 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + 2^(1/3)*(-a^4/d^3)^(1/3)*d)/a) + (a^ 
4/d^3)^(1/3)*log((2^(1/3)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d 
*x + 2/3*I*c) - (a^4/d^3)^(1/3)*d)/a)
 

Sympy [F]

\[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}} \cot {\left (c + d x \right )}\, dx \] Input:

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))**(4/3),x)
 

Output:

Integral((I*a*(tan(c + d*x) - I))**(4/3)*cot(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.92 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 2 \, \sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2^{\frac {1}{3}} a^{\frac {4}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 2 \cdot 2^{\frac {1}{3}} a^{\frac {4}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - a^{\frac {4}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {4}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{2 \, d} \] Input:

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.93 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i \, {\left (2 i \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 2 i \, \sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + i \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 2 i \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - i \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 i \, a^{\frac {1}{3}} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )\right )} a}{2 \, d} \] Input:

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))^(4/3),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/2*I*(2*I*sqrt(3)*2^(1/3)*a^(1/3)*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)) - 2*I*sqrt(3)*a^(1/3)*arc 
tan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3)) + I*2^ 
(1/3)*a^(1/3)*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)) - 2*I*2^(1/3)*a^(1/3)*log(-2^(1/3)* 
a^(1/3) + (I*a*tan(d*x + c) + a)^(1/3)) - I*a^(1/3)*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)) + 2*I*a^(1/3) 
*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3)))*a/d
 

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.45 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\ln \left (-d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}\right )\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+\ln \left (a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+2^{1/3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}\right )\,{\left (-\frac {2\,a^4}{d^3}\right )}^{1/3}+\frac {\ln \left (d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-\sqrt {3}\,d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a^4}{d^3}\right )}^{1/3}}{2}-\frac {\ln \left (d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}+2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\sqrt {3}\,d\,{\left (\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {a^4}{d^3}\right )}^{1/3}}{2}-\ln \left (2^{1/3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}-2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,a^4}{d^3}\right )}^{1/3}+\ln \left (2\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,d\,{\left (-\frac {a^4}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2\,a^4}{d^3}\right )}^{1/3} \] Input:

int(cot(c + d*x)*(a + a*tan(c + d*x)*1i)^(4/3),x)
 

Output:

log(a*(a*(tan(c + d*x)*1i + 1))^(1/3) - d*(a^4/d^3)^(1/3))*(a^4/d^3)^(1/3) 
 + log(a*(a*(tan(c + d*x)*1i + 1))^(1/3) + 2^(1/3)*d*(-a^4/d^3)^(1/3))*(-( 
2*a^4)/d^3)^(1/3) + (log(d*(a^4/d^3)^(1/3) + 2*a*(a*(tan(c + d*x)*1i + 1)) 
^(1/3) - 3^(1/2)*d*(a^4/d^3)^(1/3)*1i)*(3^(1/2)*1i - 1)*(a^4/d^3)^(1/3))/2 
 - (log(d*(a^4/d^3)^(1/3) + 2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) + 3^(1/2)* 
d*(a^4/d^3)^(1/3)*1i)*(3^(1/2)*1i + 1)*(a^4/d^3)^(1/3))/2 - log(2^(1/3)*d* 
(-a^4/d^3)^(1/3) - 2*a*(a*(tan(c + d*x)*1i + 1))^(1/3) + 2^(1/3)*3^(1/2)*d 
*(-a^4/d^3)^(1/3)*1i)*((3^(1/2)*1i)/2 + 1/2)*(-(2*a^4)/d^3)^(1/3) + log(2* 
a*(a*(tan(c + d*x)*1i + 1))^(1/3) - 2^(1/3)*d*(-a^4/d^3)^(1/3) + 2^(1/3)*3 
^(1/2)*d*(-a^4/d^3)^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(-(2*a^4)/d^3)^(1/3)
 

Reduce [F]

\[ \int \cot (c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=a^{\frac {4}{3}} \left (\left (\int \left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}} \cot \left (d x +c \right ) \tan \left (d x +c \right )d x \right ) i +\int \left (\tan \left (d x +c \right ) i +1\right )^{\frac {1}{3}} \cot \left (d x +c \right )d x \right ) \] Input:

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^(4/3),x)
 

Output:

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