Integrand size = 15, antiderivative size = 54 \[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^2}{2}-x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{b d n},1+\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(193\) vs. \(2(54)=108\).
Time = 7.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 3.57 \[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^2 \left (e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{b d n},2+\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(1+b d n) \left (\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+\operatorname {Hypergeometric2F1}\left (1,\frac {1}{b d n},1+\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text {csch}\left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sinh (b d n \log (x))\right )\right )}{2+2 b d n} \]
-((x^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)*(Coth[d*(a + b*Log[c *x^n])] - Coth[d*(a - b*n*Log[x] + b*Log[c*x^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 + 2*b*d*n))
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.67, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, 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.
\(\displaystyle \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 6074 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \int \left (c x^n\right )^{\frac {2}{n}-1} \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 6072 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \int \frac {\left (c x^n\right )^{\frac {2}{n}-1} \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {1}{2} n \left (c x^n\right )^{2/n}-2 \int \frac {\left (c x^n\right )^{\frac {2}{n}-1}}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )\right )}{n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {1}{2} n \left (c x^n\right )^{2/n}-n \left (c x^n\right )^{2/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{b d n},1+\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )\right )}{n}\) |
(x^2*((n*(c*x^n)^(2/n))/2 - n*(c*x^n)^(2/n)*Hypergeometric2F1[1, 1/(b*d*n) , 1 + 1/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)]))/(n*(c*x^n)^(2/n))
3.2.79.3.1 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])
\[\int x \coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
\[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x \coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
1/2*x^2 - integrate(x/(c^(b*d)*e^(b*d*log(x^n) + a*d) + 1), x) + integrate (x/(c^(b*d)*e^(b*d*log(x^n) + a*d) - 1), x)
\[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x\,\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]