\(\int (e x)^m \cot ^3(d (a+b \log (c x^n))) \, dx\) [228]

Optimal result

Integrand size = 21, antiderivative size = 350 \[ \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(i (1+m)-b d n) (1+m+2 i b d n) (e x)^{1+m}}{2 b^2 d^2 e (1+m) n^2}+\frac {(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}{2 b d e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}+\frac {i e^{-2 i a d} (e x)^{1+m} \left (\frac {e^{2 i a d} (1+m-2 i b d n)}{n}+\frac {e^{4 i a d} (1+m+2 i b d n) \left (c x^n\right )^{2 i b d}}{n}\right )}{2 b^2 d^2 e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i \left (1+2 m+m^2-2 b^2 d^2 n^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b^2 d^2 e (1+m) n^2} \] Output:

1/2*(I*(1+m)-b*d*n)*(1+m+2*I*b*d*n)*(e*x)^(1+m)/b^2/d^2/e/(1+m)/n^2+1/2*(e 
*x)^(1+m)*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^2/b/d/e/n/(1-exp(2*I*a*d)*(c* 
x^n)^(2*I*b*d))^2+1/2*I*(e*x)^(1+m)*(exp(2*I*a*d)*(1+m-2*I*b*d*n)/n+exp(4* 
I*a*d)*(1+m+2*I*b*d*n)*(c*x^n)^(2*I*b*d)/n)/b^2/d^2/e/exp(2*I*a*d)/n/(1-ex 
p(2*I*a*d)*(c*x^n)^(2*I*b*d))-I*(-2*b^2*d^2*n^2+m^2+2*m+1)*(e*x)^(1+m)*hyp 
ergeom([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^2/d^2/e/(1+m)/n^2
 

Mathematica [A] (verified)

Time = 13.84 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.83 \[ \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x (e x)^m \cot \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{1+m}-\frac {x (e x)^m \csc ^2\left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{2 b d n}+\frac {(1+m) x (e x)^m \csc \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \csc \left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \sin (b d n \log (x))}{2 b^2 d^2 n^2}+\frac {\left (-1-2 m-m^2+2 b^2 d^2 n^2\right ) x^{-m} (e x)^m \csc \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \left (\frac {x^{1+m} \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))}{1+m}-\frac {i e^{-\frac {(1+2 m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (i e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-e^{\frac {a (1+2 m+2 i b d n)}{b n}+(1+m+2 i b d n) \log (x)+\frac {(1+2 m+2 i b d n) \left (-n \log (x)+\log \left (c x^n\right )\right )}{n}} (1+m) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m+2 i b d n)}{2 b d n},-\frac {i (1+m+4 i b d n)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right ) \sin \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{(1+m) (1+m+2 i b d n)}\right )}{2 b^2 d^2 n^2} \] Input:

Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^3,x]
 

Output:

-((x*(e*x)^m*Cot[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(1 + m)) - (x*(e*x 
)^m*Csc[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]^2)/(2*b*d*n) 
+ ((1 + m)*x*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Csc[b*d*n*L 
og[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sin[b*d*n*Log[x]])/(2*b^2*d^ 
2*n^2) + ((-1 - 2*m - m^2 + 2*b^2*d^2*n^2)*(e*x)^m*Csc[d*(a + b*(-(n*Log[x 
]) + Log[c*x^n]))]*((x^(1 + m)*Csc[d*(a + b*Log[c*x^n])]*Sin[b*d*n*Log[x]] 
)/(1 + m) - (I*(I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Lo 
g[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n]) 
] - E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + 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^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) + (1 + m + (2*I)*b*d*n)*Log[x] + ((1 
+ 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric 
2F1[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]))])*Sin[d*(a + b*(-(n*Log[x] 
) + Log[c*x^n]))])/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n 
))*(1 + m)*(1 + m + (2*I)*b*d*n))))/(2*b^2*d^2*n^2*x^m)
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5009, 5007, 1004, 27, 1064, 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.

\(\displaystyle \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 5009

\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\)

\(\Big \downarrow \) 5007

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

\(\Big \downarrow \) 1004

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1064

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 888

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

Input:

Int[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^3,x]
 

Output:

((e*x)^(1 + m)*(((c*x^n)^((1 + m)/n)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d 
))^2)/(2*b*d*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^2) - (((-I)*(c*x^n)^( 
(1 + m)/n)*((E^((2*I)*a*d)*(1 + m - (2*I)*b*d*n))/n + (E^((4*I)*a*d)*(1 + 
m + (2*I)*b*d*n)*(c*x^n)^((2*I)*b*d))/n))/(b*d*(1 - E^((2*I)*a*d)*(c*x^n)^ 
((2*I)*b*d))) + (I*(-((E^((4*I)*a*d)*(1 + m + I*b*d*n)*(1 + m + (2*I)*b*d* 
n)*(c*x^n)^((1 + m)/n))/((1 + m)*n)) + (2*E^((4*I)*a*d)*(1 + 2*m + m^2 - 2 
*b^2*d^2*n^2)*(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)])/ 
((1 + m)*n)))/(b*d*E^((2*I)*a*d)))/(2*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 1064

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

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]

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

Input:

int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^3,x)
 

Output:

int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^3,x)
 

Fricas [F]

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

integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^3,x, algorithm="fricas")
 

Output:

integral((e*x)^m*cot(b*d*log(c*x^n) + a*d)^3, x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:

integrate((e*x)**m*cot(d*(a+b*ln(c*x**n)))**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^3,x, algorithm="maxima")
 

Output:

(4*(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 + 4*(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 - (2*b*d*e^m*n*cos(2*b*d*lo 
g(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*l 
og(x^n) + 2*a*d) + (2*b*d*e^m*n*sin(2*b*d*log(c)) + e^m*m*cos(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) + (((cos(2*b* 
d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m - 
 2*(b*d*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b*d*sin(4*b*d*log(c))*sin(2* 
b*d*log(c)))*e^m*n + (cos(2*b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log( 
c))*sin(2*b*d*log(c)))*e^m)*x*x^m*cos(2*b*d*log(x^n) + 2*a*d) - ((cos(4*b* 
d*log(c))*cos(2*b*d*log(c)) + sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + 
 2*(b*d*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b*d*cos(4*b*d*log(c))*sin(2* 
b*d*log(c)))*e^m*n + (cos(4*b*d*log(c))*cos(2*b*d*log(c)) + sin(4*b*d*log( 
c))*sin(2*b*d*log(c)))*e^m)*x*x^m*sin(2*b*d*log(x^n) + 2*a*d) - (e^m*m*sin 
(4*b*d*log(c)) + e^m*sin(4*b*d*log(c)))*x*x^m)*cos(4*b*d*log(x^n) + 4*a*d) 
 - 2*(2*b^6*d^6*e^m*n^6 - (b^4*d^4*e^m*m^2 + 2*b^4*d^4*e^m*m + b^4*d^4*e^m 
)*n^4 + (2*(b^6*d^6*cos(4*b*d*log(c))^2 + b^6*d^6*sin(4*b*d*log(c))^2)*e^m 
*n^6 - ((b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m^ 
2 + 2*(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m + 
(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m)*n^4)*c...
 

Giac [F(-1)]

Timed out. \[ \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:

integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(cot(d*(a + b*log(c*x^n)))^3*(e*x)^m,x)
 

Output:

int(cot(d*(a + b*log(c*x^n)))^3*(e*x)^m, x)
 

Reduce [F]

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

int((e*x)^m*cot(d*(a+b*log(c*x^n)))^3,x)
 

Output:

e**m*int(x**m*cot(log(x**n*c)*b*d + a*d)**3,x)