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

Optimal result

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

[Out]

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-exp(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)*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^2/d^2/e/(1+m)/n^2

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4594, 4592, 516, 608, 470, 371} \[ \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {i (e x)^{m+1} \left (-2 b^2 d^2 n^2+m^2+2 m+1\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (m+1)}{2 b d n},1-\frac {i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b^2 d^2 e (m+1) n^2}+\frac {i e^{-2 i a d} (e x)^{m+1} \left (\frac {e^{4 i a d} (2 i b d n+m+1) \left (c x^n\right )^{2 i b d}}{n}+\frac {e^{2 i a d} (-2 i b d n+m+1)}{n}\right )}{2 b^2 d^2 e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {(e x)^{m+1} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}{2 b d e n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}+\frac {(e x)^{m+1} (-b d n+i (m+1)) (2 i b d n+m+1)}{2 b^2 d^2 e (m+1) n^2} \]

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

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] - Dist[(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 516

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] + Dist[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] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 608

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] + Dis
t[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 4592

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 4594

Int[Cot[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \cot ^3(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n} \\ & = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}} \left (-i-i e^{2 i a d} x^{2 i b d}\right )^3}{\left (1-e^{2 i a d} x^{2 i b d}\right )^3} \, dx,x,c x^n\right )}{e n} \\ & = \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 {\left (i e^{-2 i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}} \left (-i-i e^{2 i a d} x^{2 i b d}\right ) \left (\frac {2 e^{2 i a d} (1+m-2 i b d n)}{n}+\frac {2 e^{4 i a d} (1+m+2 i b d n) x^{2 i b d}}{n}\right )}{\left (1-e^{2 i a d} x^{2 i b d}\right )^2} \, dx,x,c x^n\right )}{4 b d e n} \\ & = \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 {\left (e^{-4 i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}} \left (\frac {4 e^{4 i a d} (1+m-2 i b d n) (i+i m+b d n)}{n^2}+\frac {4 e^{6 i a d} (i (1+m)-b d n) (1+m+2 i b d n) x^{2 i b d}}{n^2}\right )}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{8 b^2 d^2 e n} \\ & = \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 {\left (i \left (1+2 m+m^2-2 b^2 d^2 n^2\right ) (e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{b^2 d^2 e n^3} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.56 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.83 \[ \int (e x)^m \cot ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x (e x)^m \cot \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{1+m}-\frac {x (e x)^m \csc ^2\left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{2 b d n}+\frac {(1+m) x (e x)^m \csc \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \csc \left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \sin (b d n \log (x))}{2 b^2 d^2 n^2}+\frac {\left (-1-2 m-m^2+2 b^2 d^2 n^2\right ) x^{-m} (e x)^m \csc \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \left (\frac {x^{1+m} \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))}{1+m}-\frac {i e^{-\frac {(1+2 m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (i e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-e^{\frac {a (1+2 m+2 i b d n)}{b n}+(1+m+2 i b d n) \log (x)+\frac {(1+2 m+2 i b d n) \left (-n \log (x)+\log \left (c x^n\right )\right )}{n}} (1+m) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m+2 i b d n)}{2 b d n},-\frac {i (1+m+4 i b d n)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right ) \sin \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{(1+m) (1+m+2 i b d n)}\right )}{2 b^2 d^2 n^2} \]

[In]

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

[Out]

-((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
*Log[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*Log[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)*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]))])*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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

(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*
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*si
n(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)*cos(4*b*d*log(x^n) + 4*a*d)^2 + 4*(2*(b^6*d^6*cos(2*b*d*log(c))^
2 + b^6*d^6*sin(2*b*d*log(c))^2)*e^m*n^6 - ((b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^
2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d
^4*sin(2*b*d*log(c))^2)*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d)^2 + (2*(b^6*d^6*cos(4*b*d*log(c))^2 + b^6*d^6*si
n(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*l
og(c))^2)*e^m)*n^4)*sin(4*b*d*log(x^n) + 4*a*d)^2 + 4*(2*(b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(
c))^2)*e^m*n^6 - ((b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*l
og(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^
m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d)^2 + 2*(2*b^6*d^6*e^m*n^6*cos(4*b*d*log(c)) - (b^4*d^4*e^m*m^2*cos(4*b*d*lo
g(c)) + 2*b^4*d^4*e^m*m*cos(4*b*d*log(c)) + b^4*d^4*e^m*cos(4*b*d*log(c)))*n^4 - 2*(2*(b^6*d^6*cos(4*b*d*log(c
))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 - ((b^4*d^4*cos(4*b*d*log(c))*cos(
2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*
log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) +
b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*(2*(b^6*d^6*cos(2*b*d*l
og(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 - ((b^4*d^4*cos(2*b*d*log(c))*
sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*
b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)
) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*cos(4*b*d*log(x^n) + 4
*a*d) - 4*(2*b^6*d^6*e^m*n^6*cos(2*b*d*log(c)) - (b^4*d^4*e^m*m^2*cos(2*b*d*log(c)) + 2*b^4*d^4*e^m*m*cos(2*b*
d*log(c)) + b^4*d^4*e^m*cos(2*b*d*log(c)))*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*(2*b^6*d^6*e^m*n^6*sin(4*b*d*l
og(c)) - (b^4*d^4*e^m*m^2*sin(4*b*d*log(c)) + 2*b^4*d^4*e^m*m*sin(4*b*d*log(c)) + b^4*d^4*e^m*sin(4*b*d*log(c)
))*n^4 - 2*(2*(b^6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*
n^6 - ((b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2
*(b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*
cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*cos(2*b*d*log(x^n
) + 2*a*d) + 2*(2*(b^6*d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*
e^m*n^6 - ((b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2
+ 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*
d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*sin(2*b*d*log
(x^n) + 2*a*d))*sin(4*b*d*log(x^n) + 4*a*d) + 4*(2*b^6*d^6*e^m*n^6*sin(2*b*d*log(c)) - (b^4*d^4*e^m*m^2*sin(2*
b*d*log(c)) + 2*b^4*d^4*e^m*m*sin(2*b*d*log(c)) + b^4*d^4*e^m*sin(2*b*d*log(c)))*n^4)*sin(2*b*d*log(x^n) + 2*a
*d))*integrate(1/4*(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^4*d^4*n^4*cos(b*d*log(c))*cos(b*d*log(x^n) + a*d) - 2*b^4*d^4*n^4*sin(b*d*log(c))*sin(b*d*log(x^n) + a*d
) + b^4*d^4*n^4 + (b^4*d^4*cos(b*d*log(c))^2 + b^4*d^4*sin(b*d*log(c))^2)*n^4*cos(b*d*log(x^n) + a*d)^2 + (b^4
*d^4*cos(b*d*log(c))^2 + b^4*d^4*sin(b*d*log(c))^2)*n^4*sin(b*d*log(x^n) + a*d)^2), x) + 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)*cos(4*b*d*log(x^n) + 4*a*d)^2 + 4*(2*(b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(c))^2)*e
^m*n^6 - ((b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2
+ b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m)*n^4)*
cos(2*b*d*log(x^n) + 2*a*d)^2 + (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)*sin(4*b*d*log(
x^n) + 4*a*d)^2 + 4*(2*(b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(c))^2)*e^m*n^6 - ((b^4*d^4*cos(2*b
*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c
))^2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d
)^2 + 2*(2*b^6*d^6*e^m*n^6*cos(4*b*d*log(c)) - (b^4*d^4*e^m*m^2*cos(4*b*d*log(c)) + 2*b^4*d^4*e^m*m*cos(4*b*d*
log(c)) + b^4*d^4*e^m*cos(4*b*d*log(c)))*n^4 - 2*(2*(b^6*d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin
(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 - ((b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*
log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c)
)*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*
d*log(c)))*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*(2*(b^6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6
*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 - ((b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*
b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*lo
g(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(
2*b*d*log(c)))*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*cos(4*b*d*log(x^n) + 4*a*d) - 4*(2*b^6*d^6*e^m*n^6*cos(2
*b*d*log(c)) - (b^4*d^4*e^m*m^2*cos(2*b*d*log(c)) + 2*b^4*d^4*e^m*m*cos(2*b*d*log(c)) + b^4*d^4*e^m*cos(2*b*d*
log(c)))*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*(2*b^6*d^6*e^m*n^6*sin(4*b*d*log(c)) - (b^4*d^4*e^m*m^2*sin(4*b*
d*log(c)) + 2*b^4*d^4*e^m*m*sin(4*b*d*log(c)) + b^4*d^4*e^m*sin(4*b*d*log(c)))*n^4 - 2*(2*(b^6*d^6*cos(2*b*d*l
og(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 - ((b^4*d^4*cos(2*b*d*log(c))*
sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*
b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)
) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d) + 2*(2*(b^6*d^6*cos(4*b
*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 - ((b^4*d^4*cos(4*b*d*log(
c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))*co
s(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*lo
g(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*sin(4*b*d*log(x^n)
+ 4*a*d) + 4*(2*b^6*d^6*e^m*n^6*sin(2*b*d*log(c)) - (b^4*d^4*e^m*m^2*sin(2*b*d*log(c)) + 2*b^4*d^4*e^m*m*sin(
2*b*d*log(c)) + b^4*d^4*e^m*sin(2*b*d*log(c)))*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*integrate(-1/4*(x^m*cos(b*d*l
og(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^4*d^4*n^4*cos(b*d*log(c))*c
os(b*d*log(x^n) + a*d) - 2*b^4*d^4*n^4*sin(b*d*log(c))*sin(b*d*log(x^n) + a*d) - b^4*d^4*n^4 - (b^4*d^4*cos(b*
d*log(c))^2 + b^4*d^4*sin(b*d*log(c))^2)*n^4*cos(b*d*log(x^n) + a*d)^2 - (b^4*d^4*cos(b*d*log(c))^2 + b^4*d^4*
sin(b*d*log(c))^2)*n^4*sin(b*d*log(x^n) + a*d)^2), x) + (((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*l
og(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*cos(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*sin(2*b*d*log(x^n) + 2*a*d) - (e^m*m*cos(
4*b*d*log(c)) + e^m*cos(4*b*d*log(c)))*x*x^m)*sin(4*b*d*log(x^n) + 4*a*d))/(4*b^2*d^2*n^2*cos(2*b*d*log(c))*co
s(2*b*d*log(x^n) + 2*a*d) - 4*b^2*d^2*n^2*sin(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d) - b^2*d^2*n^2 - (b^2*d
^2*cos(4*b*d*log(c))^2 + b^2*d^2*sin(4*b*d*log(c))^2)*n^2*cos(4*b*d*log(x^n) + 4*a*d)^2 - 4*(b^2*d^2*cos(2*b*d
*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*cos(2*b*d*log(x^n) + 2*a*d)^2 - (b^2*d^2*cos(4*b*d*log(c))^2 + b
^2*d^2*sin(4*b*d*log(c))^2)*n^2*sin(4*b*d*log(x^n) + 4*a*d)^2 - 4*(b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2
*b*d*log(c))^2)*n^2*sin(2*b*d*log(x^n) + 2*a*d)^2 - 2*(b^2*d^2*n^2*cos(4*b*d*log(c)) - 2*(b^2*d^2*cos(4*b*d*lo
g(c))*cos(2*b*d*log(c)) + b^2*d^2*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*cos(2*b*d*log(x^n) + 2*a*d) - 2*(b^
2*d^2*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^2*d^2*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*sin(2*b*d*log(x^n
) + 2*a*d))*cos(4*b*d*log(x^n) + 4*a*d) + 2*(b^2*d^2*n^2*sin(4*b*d*log(c)) - 2*(b^2*d^2*cos(2*b*d*log(c))*sin(
4*b*d*log(c)) - b^2*d^2*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*cos(2*b*d*log(x^n) + 2*a*d) + 2*(b^2*d^2*cos(
4*b*d*log(c))*cos(2*b*d*log(c)) + b^2*d^2*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*sin(2*b*d*log(x^n) + 2*a*d)
)*sin(4*b*d*log(x^n) + 4*a*d))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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