Integrand size = 15, antiderivative size = 177 \[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=\frac {(3+m) (5+m) (e x)^{1+m}}{8 e (1+m)}-\frac {(e x)^{1+m} \left (1+e^{2 a} x^4\right )^2}{4 e \left (1-e^{2 a} x^4\right )^2}-\frac {e^{-2 a} (e x)^{1+m} \left (e^{2 a} (3-m)-e^{4 a} (5+m) x^4\right )}{8 e \left (1-e^{2 a} x^4\right )}-\frac {\left (9+2 m+m^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},e^{2 a} x^4\right )}{4 e (1+m)} \]
1/8*(3+m)*(5+m)*(e*x)^(1+m)/e/(1+m)-1/4*(e*x)^(1+m)*(1+exp(2*a)*x^4)^2/e/( 1-exp(2*a)*x^4)^2-1/8*(e*x)^(1+m)*(exp(2*a)*(3-m)-exp(4*a)*(5+m)*x^4)/e/ex p(2*a)/(1-exp(2*a)*x^4)-1/4*(m^2+2*m+9)*(e*x)^(1+m)*hypergeom([1, 1/4+1/4* m],[5/4+1/4*m],exp(2*a)*x^4)/e/(1+m)
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=-\frac {x (e x)^m \left (-1+6 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},x^4 (\cosh (2 a)+\sinh (2 a))\right )-12 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{4},\frac {5+m}{4},x^4 (\cosh (2 a)+\sinh (2 a))\right )+8 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{4},\frac {5+m}{4},x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{1+m} \]
-((x*(e*x)^m*(-1 + 6*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, x^4*(Cosh[ 2*a] + Sinh[2*a])] - 12*Hypergeometric2F1[2, (1 + m)/4, (5 + m)/4, x^4*(Co sh[2*a] + Sinh[2*a])] + 8*Hypergeometric2F1[3, (1 + m)/4, (5 + m)/4, x^4*( Cosh[2*a] + Sinh[2*a])]))/(1 + m))
Time = 0.46 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6072, 968, 27, 1047, 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.
| \(\displaystyle \int (e x)^m \coth ^3(a+2 \log (x)) \, dx\) |
| \(\Big \downarrow \) 6072 |
| \(\displaystyle \int \frac {\left (-e^{2 a} x^4-1\right )^3 (e x)^m}{\left (1-e^{2 a} x^4\right )^3}dx\) |
| \(\Big \downarrow \) 968 |
| \(\displaystyle \frac {1}{8} e^{-2 a} \int -\frac {2 (e x)^m \left (e^{2 a} x^4+1\right ) \left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right )}{\left (1-e^{2 a} x^4\right )^2}dx-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} e^{-2 a} \int \frac {(e x)^m \left (e^{2 a} x^4+1\right ) \left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right )}{\left (1-e^{2 a} x^4\right )^2}dx-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}\) |
| \(\Big \downarrow \) 1047 |
| \(\displaystyle -\frac {1}{4} e^{-2 a} \left (\frac {1}{4} e^{-2 a} \int \frac {2 (e x)^m \left (e^{6 a} (m+3) (m+5) x^4+e^{4 a} (1-m) (3-m)\right )}{1-e^{2 a} x^4}dx+\frac {\left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right ) (e x)^{m+1}}{2 e \left (1-e^{2 a} x^4\right )}\right )-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} e^{-2 a} \left (\frac {1}{2} e^{-2 a} \int \frac {(e x)^m \left (e^{6 a} (m+3) (m+5) x^4+e^{4 a} (1-m) (3-m)\right )}{1-e^{2 a} x^4}dx+\frac {\left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right ) (e x)^{m+1}}{2 e \left (1-e^{2 a} x^4\right )}\right )-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle -\frac {1}{4} e^{-2 a} \left (\frac {1}{2} e^{-2 a} \left (2 e^{4 a} \left (m^2+2 m+9\right ) \int \frac {(e x)^m}{1-e^{2 a} x^4}dx-\frac {e^{4 a} (m+3) (m+5) (e x)^{m+1}}{e (m+1)}\right )+\frac {\left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right ) (e x)^{m+1}}{2 e \left (1-e^{2 a} x^4\right )}\right )-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle -\frac {1}{4} e^{-2 a} \left (\frac {1}{2} e^{-2 a} \left (\frac {2 e^{4 a} \left (m^2+2 m+9\right ) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},e^{2 a} x^4\right )}{e (m+1)}-\frac {e^{4 a} (m+3) (m+5) (e x)^{m+1}}{e (m+1)}\right )+\frac {\left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right ) (e x)^{m+1}}{2 e \left (1-e^{2 a} x^4\right )}\right )-\frac {\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}\) |
-1/4*((e*x)^(1 + m)*(1 + E^(2*a)*x^4)^2)/(e*(1 - E^(2*a)*x^4)^2) - (((e*x) ^(1 + m)*(E^(2*a)*(3 - m) - E^(4*a)*(5 + m)*x^4))/(2*e*(1 - E^(2*a)*x^4)) + (-((E^(4*a)*(3 + m)*(5 + m)*(e*x)^(1 + m))/(e*(1 + m))) + (2*E^(4*a)*(9 + 2*m + m^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, E^(2 *a)*x^4])/(e*(1 + m)))/(2*E^(2*a)))/(4*E^(2*a))
3.2.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
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}, x] && IGtQ[n , 0] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
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]
\[\int \left (e x \right )^{m} \coth \left (a +2 \ln \left (x \right )\right )^{3}d x\]
\[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )^{3} \,d x } \]
\[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=\int \left (e x\right )^{m} \coth ^{3}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \]
\[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )^{3} \,d x } \]
\[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )^{3} \,d x } \]
Timed out. \[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx=\int {\mathrm {coth}\left (a+2\,\ln \left (x\right )\right )}^3\,{\left (e\,x\right )}^m \,d x \]