Integrand size = 17, antiderivative size = 55 \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {1}{2 x^2}+\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x^2} \]
Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(55)=110\).
Time = 4.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.47 \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )-\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{b d n},2-\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{-1+b d n}+\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text {csch}\left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sinh (b d n \log (x))}{2 x^2} \]
(Coth[d*(a + b*Log[c*x^n])] - Coth[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (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) + Hypergeometric2F1[1, -(1/(b *d*n)), 1 - 1/(b*d*n), E^(2*d*(a + b*Log[c*x^n]))] + Csch[d*(a + b*Log[c*x ^n])]*Csch[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sinh[b*d*n*Log[x]])/(2*x^2)
Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.65, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, 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.
\(\displaystyle \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 6074 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \left (c x^n\right )^{-1-\frac {2}{n}} \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n x^2}\) |
\(\Big \downarrow \) 6072 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}} \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{n x^2}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (-2 \int \frac {\left (c x^n\right )^{-1-\frac {2}{n}}}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )-\frac {1}{2} n \left (c x^n\right )^{-2/n}\right )}{n x^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (n \left (c x^n\right )^{-2/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )-\frac {1}{2} n \left (c x^n\right )^{-2/n}\right )}{n x^2}\) |
((c*x^n)^(2/n)*(-1/2*n/(c*x^n)^(2/n) + (n*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)))/(n*x^2)
3.2.83.3.1 Defintions of rubi rules used
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[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])
\[\int \frac {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
-1/2/x^2 - integrate(1/(c^(b*d)*x^3*e^(b*d*log(x^n) + a*d) + x^3), x) + in tegrate(1/(c^(b*d)*x^3*e^(b*d*log(x^n) + a*d) - x^3), x)
\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]