\(\int (e x)^m \coth (d (a+b \log (c x^n))) \, dx\) [199]

Optimal result

Integrand size = 19, antiderivative size = 87 \[ \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m}}{e (1+m)}-\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2 b d n},1+\frac {1+m}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 8.95 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.82 \[ \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m \left (-\operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2 b d n},1+\frac {1+m}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-\frac {e^{2 a d} (1+m) \left (c x^n\right )^{2 b d} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+2 b d n}{2 b d n},\frac {1+m+4 b d n}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{1+m+2 b d n}\right )}{1+m} \] Input:

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

Output:

(x*(e*x)^m*(-Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n) 
, E^(2*d*(a + b*Log[c*x^n]))] - (E^(2*a*d)*(1 + m)*(c*x^n)^(2*b*d)*Hyperge 
ometric2F1[1, (1 + m + 2*b*d*n)/(2*b*d*n), (1 + m + 4*b*d*n)/(2*b*d*n), E^ 
(2*a*d)*(c*x^n)^(2*b*d)])/(1 + m + 2*b*d*n)))/(1 + m)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6074, 6072, 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.

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

\(\Big \downarrow \) 6074

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

\(\Big \downarrow \) 6072

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

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 6072

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

rule 6074

Int[Coth[((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)*Coth[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

e^m*x*x^m/(m + 1) - e^m*integrate(x^m/(c^(b*d)*e^(b*d*log(x^n) + a*d) + 1) 
, x) + e^m*integrate(x^m/(c^(b*d)*e^(b*d*log(x^n) + a*d) - 1), x)
 

Giac [F]

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

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

Output:

integrate((e*x)^m*coth((b*log(c*x^n) + a)*d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {e^{m} \left (x^{m} x +2 \left (\int \frac {x^{m}}{x^{2 b d n} e^{2 a d} c^{2 b d}-1}d x \right ) m +2 \left (\int \frac {x^{m}}{x^{2 b d n} e^{2 a d} c^{2 b d}-1}d x \right )\right )}{m +1} \] Input:

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

Output:

(e**m*(x**m*x + 2*int(x**m/(x**(2*b*d*n)*e**(2*a*d)*c**(2*b*d) - 1),x)*m + 
 2*int(x**m/(x**(2*b*d*n)*e**(2*a*d)*c**(2*b*d) - 1),x)))/(m + 1)