\(\int (e x)^m \cot ^2(d (a+b \log (c x^n))) \, dx\) [227]

Optimal result

Integrand size = 21, antiderivative size = 195 \[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(i (1+m)-b d n) (e x)^{1+m}}{b d e (1+m) n}+\frac {i (e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d e n} \] Output:

(I*(1+m)-b*d*n)*(e*x)^(1+m)/b/d/e/(1+m)/n+I*(e*x)^(1+m)*(1+exp(2*I*a*d)*(c 
*x^n)^(2*I*b*d))/b/d/e/n/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))-2*I*(e*x)^(1+m 
)*hypergeom([1, -1/2*I*(1+m)/b/d/n],[1-1/2*I*(1+m)/b/d/n],exp(2*I*a*d)*(c* 
x^n)^(2*I*b*d))/b/d/e/n
 

Mathematica [A] (verified)

Time = 13.40 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.77 \[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=(e x)^m \left (-\frac {x}{1+m}+\frac {i e^{-\frac {(1+2 m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-2 m} \left (i e^{\frac {(1+2 m) \left (a+b \log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{\frac {(1+2 m) \left (a+b \log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-e^{\frac {(1+2 m+2 i b d n) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} (1+m) x^{1+2 m+2 i b d n} \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m+2 i b d n)}{2 b d n},-\frac {i (1+m+4 i b d n)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b d n (1+m+2 i b d n)}\right ) \] Input:

Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2,x]
 

Output:

(e*x)^m*(-(x/(1 + m)) + (I*(I*E^(((1 + 2*m)*(a + b*Log[c*x^n]))/(b*n))*(1 
+ m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] - E^(((1 + 2*m)*(a + b*Log[c* 
x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m) 
)/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] - 
E^(((1 + 2*m + (2*I)*b*d*n)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m 
)*x^(1 + 2*m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)* 
b*d*n))/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*d*(a + 
 b*Log[c*x^n]))]))/(b*d*E^(((1 + 2*m)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b* 
n))*n*(1 + m + (2*I)*b*d*n)*x^(2*m)))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5009, 5007, 1004, 27, 959, 888}

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.

\(\displaystyle \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 5009

\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\)

\(\Big \downarrow \) 5007

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

\(\Big \downarrow \) 1004

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \left (\frac {i \left (c x^n\right )^{\frac {m+1}{n}} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \left (2 e^{2 i a d} \left (c x^n\right )^{\frac {m+1}{n}} \operatorname {Hypergeometric2F1}\left (1,-\frac {i (m+1)}{2 b d n},1-\frac {i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac {e^{2 i a d} (i b d n+m+1) \left (c x^n\right )^{\frac {m+1}{n}}}{m+1}\right )}{b d}\right )}{e n}\)

Input:

Int[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2,x]
 

Output:

((e*x)^(1 + m)*((I*(c*x^n)^((1 + m)/n)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b 
*d)))/(b*d*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))) - (I*(-((E^((2*I)*a*d) 
*(1 + m + I*b*d*n)*(c*x^n)^((1 + m)/n))/(1 + m)) + 2*E^((2*I)*a*d)*(c*x^n) 
^((1 + m)/n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)*( 
1 + m))/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]))/(b*d*E^((2*I)*a*d))) 
)/(e*n*(c*x^n)^((1 + m)/n))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 888

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

rule 959

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

rule 1004

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

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]
 

rule 5009

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

Maple [F]

\[\int \left (e x \right )^{m} {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]

Input:

int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^2,x)
 

Output:

int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^2,x)
 

Fricas [F]

\[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \] Input:

integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x, algorithm="fricas")
 

Output:

integral((e*x)^m*cot(b*d*log(c*x^n) + a*d)^2, x)
 

Sympy [F]

\[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \cot ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate((e*x)**m*cot(d*(a+b*ln(c*x**n)))**2,x)
 

Output:

Integral((e*x)**m*cot(a*d + b*d*log(c*x**n))**2, x)
 

Maxima [F]

\[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \] Input:

integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x, algorithm="maxima")
 

Output:

-((b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*e^m*n*x*x^m*cos(2*b* 
d*log(x^n) + 2*a*d)^2 + (b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2 
)*e^m*n*x*x^m*sin(2*b*d*log(x^n) + 2*a*d)^2 + b*d*e^m*n*x*x^m - 2*(b*d*e^m 
*n*cos(2*b*d*log(c)) - e^m*m*sin(2*b*d*log(c)) - e^m*sin(2*b*d*log(c)))*x* 
x^m*cos(2*b*d*log(x^n) + 2*a*d) + 2*(b*d*e^m*n*sin(2*b*d*log(c)) + e^m*m*c 
os(2*b*d*log(c)) + e^m*cos(2*b*d*log(c)))*x*x^m*sin(2*b*d*log(x^n) + 2*a*d 
) + (((b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m*m^2 
+ 2*(b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m*m + (b 
^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m)*n^2*cos(2*b 
*d*log(x^n) + 2*a*d)^2 + ((b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d 
*log(c))^2)*e^m*m^2 + 2*(b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*l 
og(c))^2)*e^m*m + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c)) 
^2)*e^m)*n^2*sin(2*b*d*log(x^n) + 2*a*d)^2 - 2*(b^2*d^2*e^m*m^2*cos(2*b*d* 
log(c)) + 2*b^2*d^2*e^m*m*cos(2*b*d*log(c)) + b^2*d^2*e^m*cos(2*b*d*log(c) 
))*n^2*cos(2*b*d*log(x^n) + 2*a*d) + 2*(b^2*d^2*e^m*m^2*sin(2*b*d*log(c)) 
+ 2*b^2*d^2*e^m*m*sin(2*b*d*log(c)) + b^2*d^2*e^m*sin(2*b*d*log(c)))*n^2*s 
in(2*b*d*log(x^n) + 2*a*d) + (b^2*d^2*e^m*m^2 + 2*b^2*d^2*e^m*m + b^2*d^2* 
e^m)*n^2)*integrate((x^m*cos(b*d*log(x^n) + a*d)*sin(b*d*log(c)) + x^m*cos 
(b*d*log(c))*sin(b*d*log(x^n) + a*d))/(2*b^2*d^2*n^2*cos(b*d*log(c))*cos(b 
*d*log(x^n) + a*d) - 2*b^2*d^2*n^2*sin(b*d*log(c))*sin(b*d*log(x^n) + a...
 

Giac [F(-1)]

Timed out. \[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:

integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2\,{\left (e\,x\right )}^m \,d x \] Input:

int(cot(d*(a + b*log(c*x^n)))^2*(e*x)^m,x)
 

Output:

int(cot(d*(a + b*log(c*x^n)))^2*(e*x)^m, x)
 

Reduce [F]

\[ \int (e x)^m \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=e^{m} \left (\int x^{m} {\cot \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}^{2}d x \right ) \] Input:

int((e*x)^m*cot(d*(a+b*log(c*x^n)))^2,x)
                                                                                    
                                                                                    
 

Output:

e**m*int(x**m*cot(log(x**n*c)*b*d + a*d)**2,x)