\(\int (e x)^m \cot ^3(a+i \log (x)) \, dx\) [203]

Optimal result

Integrand size = 17, antiderivative size = 127 \[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=-\frac {i x (e x)^m}{1+m}-\frac {2 i x (e x)^m}{\left (1-\frac {e^{2 i a}}{x^2}\right )^2}+\frac {i (1-m) x (e x)^m}{1-\frac {e^{2 i a}}{x^2}}+\frac {i \left (3+2 m+m^2\right ) x (e x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right )}{1+m} \] Output:

-I*x*(e*x)^m/(1+m)-2*I*x*(e*x)^m/(1-exp(2*I*a)/x^2)^2+I*(1-m)*x*(e*x)^m/(1 
-exp(2*I*a)/x^2)+I*(m^2+2*m+3)*x*(e*x)^m*hypergeom([1, -1/2-1/2*m],[1/2-1/ 
2*m],exp(2*I*a)/x^2)/(1+m)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=-\frac {i x (e x)^m \left (-1+6 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )-12 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )+8 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{2},\frac {3+m}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )\right )}{1+m} \] Input:

Integrate[(e*x)^m*Cot[a + I*Log[x]]^3,x]
 

Output:

((-I)*x*(e*x)^m*(-1 + 6*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, x^2*(Co 
s[2*a] - I*Sin[2*a])] - 12*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, x^2* 
(Cos[2*a] - I*Sin[2*a])] + 8*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, x^ 
2*(Cos[2*a] - I*Sin[2*a])]))/(1 + m)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.78, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {5007, 999, 26, 370, 27, 439, 27, 363, 278}

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.

\(\displaystyle \int (e x)^m \cot ^3(a+i \log (x)) \, dx\)

\(\Big \downarrow \) 5007

\(\displaystyle \int \frac {\left (-\frac {i e^{2 i a}}{x^2}-i\right )^3 (e x)^m}{\left (1-\frac {e^{2 i a}}{x^2}\right )^3}dx\)

\(\Big \downarrow \) 999

\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \int \frac {i \left (1+\frac {e^{2 i a}}{x^2}\right )^3 \left (\frac {1}{x}\right )^{-m-2}}{\left (1-\frac {e^{2 i a}}{x^2}\right )^3}d\frac {1}{x}\)

\(\Big \downarrow \) 26

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \int \frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^3 \left (\frac {1}{x}\right )^{-m-2}}{\left (1-\frac {e^{2 i a}}{x^2}\right )^3}d\frac {1}{x}\)

\(\Big \downarrow \) 370

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{4} e^{-2 i a} \int \frac {2 \left (1+\frac {e^{2 i a}}{x^2}\right ) \left (e^{2 i a} (m+3)-\frac {e^{4 i a} (1-m)}{x^2}\right ) \left (\frac {1}{x}\right )^{-m-2}}{\left (1-\frac {e^{2 i a}}{x^2}\right )^2}d\frac {1}{x}+\frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-1}}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} e^{-2 i a} \int \frac {\left (1+\frac {e^{2 i a}}{x^2}\right ) \left (e^{2 i a} (m+3)-\frac {e^{4 i a} (1-m)}{x^2}\right ) \left (\frac {1}{x}\right )^{-m-2}}{\left (1-\frac {e^{2 i a}}{x^2}\right )^2}d\frac {1}{x}+\frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-1}}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}\right )\)

\(\Big \downarrow \) 439

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} e^{-2 i a} \left (\frac {1}{2} e^{-2 i a} \int \frac {2 \left (e^{4 i a} (m+2) (m+3)-\frac {e^{6 i a} (1-m) m}{x^2}\right ) \left (\frac {1}{x}\right )^{-m-2}}{1-\frac {e^{2 i a}}{x^2}}d\frac {1}{x}+\frac {\left (\frac {1}{x}\right )^{-m-1} \left (e^{2 i a} (m+3)-\frac {e^{4 i a} (1-m)}{x^2}\right )}{1-\frac {e^{2 i a}}{x^2}}\right )+\frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-1}}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} e^{-2 i a} \left (e^{-2 i a} \int \frac {\left (e^{4 i a} (m+2) (m+3)-\frac {e^{6 i a} (1-m) m}{x^2}\right ) \left (\frac {1}{x}\right )^{-m-2}}{1-\frac {e^{2 i a}}{x^2}}d\frac {1}{x}+\frac {\left (\frac {1}{x}\right )^{-m-1} \left (e^{2 i a} (m+3)-\frac {e^{4 i a} (1-m)}{x^2}\right )}{1-\frac {e^{2 i a}}{x^2}}\right )+\frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-1}}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}\right )\)

\(\Big \downarrow \) 363

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} e^{-2 i a} \left (e^{-2 i a} \left (2 e^{4 i a} \left (m^2+2 m+3\right ) \int \frac {\left (\frac {1}{x}\right )^{-m-2}}{1-\frac {e^{2 i a}}{x^2}}d\frac {1}{x}-\frac {e^{4 i a} (1-m) m \left (\frac {1}{x}\right )^{-m-1}}{m+1}\right )+\frac {\left (\frac {1}{x}\right )^{-m-1} \left (e^{2 i a} (m+3)-\frac {e^{4 i a} (1-m)}{x^2}\right )}{1-\frac {e^{2 i a}}{x^2}}\right )+\frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-1}}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}\right )\)

\(\Big \downarrow \) 278

\(\displaystyle -i \left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} e^{-2 i a} \left (e^{-2 i a} \left (-\frac {2 e^{4 i a} \left (m^2+2 m+3\right ) \left (\frac {1}{x}\right )^{-m-1} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m-1),\frac {1-m}{2},\frac {e^{2 i a}}{x^2}\right )}{m+1}-\frac {e^{4 i a} (1-m) m \left (\frac {1}{x}\right )^{-m-1}}{m+1}\right )+\frac {\left (\frac {1}{x}\right )^{-m-1} \left (e^{2 i a} (m+3)-\frac {e^{4 i a} (1-m)}{x^2}\right )}{1-\frac {e^{2 i a}}{x^2}}\right )+\frac {\left (1+\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-1}}{2 \left (1-\frac {e^{2 i a}}{x^2}\right )^2}\right )\)

Input:

Int[(e*x)^m*Cot[a + I*Log[x]]^3,x]
 

Output:

(-I)*(x^(-1))^m*(e*x)^m*(((1 + E^((2*I)*a)/x^2)^2*(x^(-1))^(-1 - m))/(2*(1 
 - E^((2*I)*a)/x^2)^2) + (((E^((2*I)*a)*(3 + m) - (E^((4*I)*a)*(1 - m))/x^ 
2)*(x^(-1))^(-1 - m))/(1 - E^((2*I)*a)/x^2) + (-((E^((4*I)*a)*(1 - m)*m*(x 
^(-1))^(-1 - m))/(1 + m)) - (2*E^((4*I)*a)*(3 + 2*m + m^2)*(x^(-1))^(-1 - 
m)*Hypergeometric2F1[1, (-1 - m)/2, (1 - m)/2, E^((2*I)*a)/x^2])/(1 + m))/ 
E^((2*I)*a))/(2*E^((2*I)*a)))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 370

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + 
 d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(e*x) 
^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a 
*d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], 
 x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] 
&& GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 439

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
 + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p 
+ 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( 
p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 
))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G 
tQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
 

rule 999

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(-(e*x)^m)*(x^(-1))^m   Subst[Int[(a + b/x^n)^p*( 
(c + d/x^n)^q/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m, p, q} 
, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] &&  !RationalQ[m]
 

rule 5007

Int[Cot[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] 
:> Int[(e*x)^m*((-I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 - E^(2*I*a*d)*x^(2*I*b* 
d)))^p, x] /; FreeQ[{a, b, d, e, m, p}, x]
 

Maple [F]

\[\int \left (e x \right )^{m} \cot \left (a +i \ln \left (x \right )\right )^{3}d x\]

Input:

int((e*x)^m*cot(a+I*ln(x))^3,x)
 

Output:

int((e*x)^m*cot(a+I*ln(x))^3,x)
 

Fricas [F]

\[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )^{3} \,d x } \] Input:

integrate((e*x)^m*cot(a+I*log(x))^3,x, algorithm="fricas")
 

Output:

integral(-(-I*x^6 - 3*I*x^4*e^(2*I*a) - 3*I*x^2*e^(4*I*a) - I*e^(6*I*a))*e 
^(m*log(e) + m*log(x))/(x^6 - 3*x^4*e^(2*I*a) + 3*x^2*e^(4*I*a) - e^(6*I*a 
)), x)
 

Sympy [F]

\[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=\int \left (e x\right )^{m} \cot ^{3}{\left (a + i \log {\left (x \right )} \right )}\, dx \] Input:

integrate((e*x)**m*cot(a+I*ln(x))**3,x)
 

Output:

Integral((e*x)**m*cot(a + I*log(x))**3, x)
 

Maxima [F]

\[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )^{3} \,d x } \] Input:

integrate((e*x)^m*cot(a+I*log(x))^3,x, algorithm="maxima")
 

Output:

integrate((e*x)^m*cot(a + I*log(x))^3, x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )^{3} \,d x } \] Input:

integrate((e*x)^m*cot(a+I*log(x))^3,x, algorithm="giac")
 

Output:

integrate((e*x)^m*cot(a + I*log(x))^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=\int {\mathrm {cot}\left (a+\ln \left (x\right )\,1{}\mathrm {i}\right )}^3\,{\left (e\,x\right )}^m \,d x \] Input:

int(cot(a + log(x)*1i)^3*(e*x)^m,x)
 

Output:

int(cot(a + log(x)*1i)^3*(e*x)^m, x)
 

Reduce [F]

\[ \int (e x)^m \cot ^3(a+i \log (x)) \, dx=e^{m} \left (\int x^{m} \cot \left (\mathrm {log}\left (x \right ) i +a \right )^{3}d x \right ) \] Input:

int((e*x)^m*cot(a+I*log(x))^3,x)
 

Output:

e**m*int(x**m*cot(log(x)*i + a)**3,x)