Integrand size = 15, antiderivative size = 126 \[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\left (1+\frac {1}{b d n}\right ) x+\frac {x \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 b d n},1+\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n} \]
(1+1/b/d/n)*x+x*(1+exp(2*a*d)*(c*x^n)^(2*b*d))/b/d/n/(1-exp(2*a*d)*(c*x^n) ^(2*b*d))-2*x*hypergeom([1, 1/2/b/d/n],[1+1/2/b/d/n],exp(2*a*d)*(c*x^n)^(2 *b*d))/b/d/n
Time = 8.40 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.27 \[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x \left (-e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{2 b d n},2+\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(1+2 b d n) \left (b d n-\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 b d n},1+\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )\right )}{b d n (1+2 b d n)} \]
(x*(-(E^(2*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + 1/(2*b*d*n), 2 + 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))]) + (1 + 2*b*d*n)*(b*d*n - Coth[d *(a + b*Log[c*x^n])] - Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))])))/(b*d*n*(1 + 2*b*d*n))
Time = 0.46 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6070, 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.
| \(\displaystyle \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6070 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 6072 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{\frac {1}{n}-1} \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^2}{\left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^2}d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 1004 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {e^{-2 a d} \int -\frac {2 \left (c x^n\right )^{\frac {1}{n}-1} \left (e^{4 a d} \left (b d+\frac {1}{n}\right ) \left (c x^n\right )^{2 b d}+\frac {e^{2 a d} (1-b d n)}{n}\right )}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{2 b d}+\frac {\left (c x^n\right )^{\frac {1}{n}} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}\right )}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {\left (c x^n\right )^{\frac {1}{n}} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {e^{-2 a d} \int \frac {\left (c x^n\right )^{\frac {1}{n}-1} \left (e^{4 a d} \left (b d+\frac {1}{n}\right ) \left (c x^n\right )^{2 b d}+\frac {e^{2 a d} (1-b d n)}{n}\right )}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{b d}\right )}{n}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {\left (c x^n\right )^{\frac {1}{n}} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {e^{-2 a d} \left (\frac {2 e^{2 a d} \int \frac {\left (c x^n\right )^{\frac {1}{n}-1}}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )}{n}-e^{2 a d} (b d n+1) \left (c x^n\right )^{\frac {1}{n}}\right )}{b d}\right )}{n}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {\left (c x^n\right )^{\frac {1}{n}} \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {e^{-2 a d} \left (2 e^{2 a d} \left (c x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 b d n},1+\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )-e^{2 a d} (b d n+1) \left (c x^n\right )^{\frac {1}{n}}\right )}{b d}\right )}{n}\) |
(x*(((c*x^n)^n^(-1)*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*(1 - E^(2*a*d)*( c*x^n)^(2*b*d))) - (-(E^(2*a*d)*(1 + b*d*n)*(c*x^n)^n^(-1)) + 2*E^(2*a*d)* (c*x^n)^n^(-1)*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), E^(2*a*d )*(c*x^n)^(2*b*d)])/(b*d*E^(2*a*d))))/(n*(c*x^n)^n^(-1))
3.2.87.3.1 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[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S imp[x/(n*(c*x^n)^(1/n)) Subst[Int[x^(1/n - 1)*Coth[d*(a + b*Log[x])]^p, x ], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1] )
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]
\[\int {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]
\[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
\[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \coth ^{2}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
\[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
(b*c^(2*b*d)*d*n*x*e^(2*b*d*log(x^n) + 2*a*d) - (b*d*n + 2)*x)/(b*c^(2*b*d )*d*n*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n) - integrate(1/(b*c^(b*d)*d*n*e^( b*d*log(x^n) + a*d) + b*d*n), x) + integrate(1/(b*c^(b*d)*d*n*e^(b*d*log(x ^n) + a*d) - b*d*n), x)
\[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
Timed out. \[ \int \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]