Integrand size = 19, antiderivative size = 135 \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {2-b d n}{2 b d n x^2}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x^2 \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x^2} \]
1/2*(-b*d*n+2)/b/d/n/x^2+(1+exp(2*a*d)*(c*x^n)^(2*b*d))/b/d/n/x^2/(1-exp(2 *a*d)*(c*x^n)^(2*b*d))-2*hypergeom([1, -1/b/d/n],[1-1/b/d/n],exp(2*a*d)*(c *x^n)^(2*b*d))/b/d/n/x^2
Time = 3.64 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.16 \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{b d n},2-\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(-1+b d n) \left (b d n+2 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{2 b d n (-1+b d n) x^2} \]
(2*E^(2*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - 1/(b*d*n), 2 - 1/(b *d*n), E^(2*d*(a + b*Log[c*x^n]))] - (-1 + b*d*n)*(b*d*n + 2*Coth[d*(a + b *Log[c*x^n])] + 2*Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), E^(2*d *(a + b*Log[c*x^n]))]))/(2*b*d*n*(-1 + b*d*n)*x^2)
Time = 0.48 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6074, 6072, 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.
\(\displaystyle \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 6074 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \left (c x^n\right )^{-1-\frac {2}{n}} \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n x^2}\) |
\(\Big \downarrow \) 6072 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}} \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^2}{\left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^2}d\left (c x^n\right )}{n x^2}\) |
\(\Big \downarrow \) 1004 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \int \frac {2 \left (c x^n\right )^{-1-\frac {2}{n}} \left (\frac {e^{4 a d} (2-b d n) \left (c x^n\right )^{2 b d}}{n}+\frac {e^{2 a d} (b d n+2)}{n}\right )}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{2 b d}+\frac {\left (c x^n\right )^{-2/n} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}\right )}{n x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}} \left (\frac {e^{4 a d} (2-b d n) \left (c x^n\right )^{2 b d}}{n}+\frac {e^{2 a d} (b d n+2)}{n}\right )}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{b d}+\frac {\left (c x^n\right )^{-2/n} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}\right )}{n x^2}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \left (\frac {4 e^{2 a d} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}}}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{n}+\frac {1}{2} e^{2 a d} (2-b d n) \left (c x^n\right )^{-2/n}\right )}{b d}+\frac {\left (c x^n\right )^{-2/n} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}\right )}{n x^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {e^{-2 a d} \left (\frac {1}{2} e^{2 a d} (2-b d n) \left (c x^n\right )^{-2/n}-2 e^{2 a d} \left (c x^n\right )^{-2/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )\right )}{b d}+\frac {\left (c x^n\right )^{-2/n} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}\right )}{n x^2}\) |
((c*x^n)^(2/n)*((1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*(c*x^n)^(2/n)*(1 - E^ (2*a*d)*(c*x^n)^(2*b*d))) + ((E^(2*a*d)*(2 - b*d*n))/(2*(c*x^n)^(2/n)) - ( 2*E^(2*a*d)*Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), E^(2*a*d)*(c *x^n)^(2*b*d)])/(c*x^n)^(2/n))/(b*d*E^(2*a*d))))/(n*x^2)
3.2.90.3.1 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, 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]
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[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])
\[\int \frac {{\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}}{x^{3}}d x\]
\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}} \,d x } \]
-1/2*(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n + 4)/(b*c^(2*b*d) *d*n*x^2*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n*x^2) + 2*integrate(1/(b*c^(b*d )*d*n*x^3*e^(b*d*log(x^n) + a*d) + b*d*n*x^3), x) - 2*integrate(1/(b*c^(b* d)*d*n*x^3*e^(b*d*log(x^n) + a*d) - b*d*n*x^3), x)
\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^3} \,d x \]