\(\int \frac {\coth ^2(d (a+b \log (c x^n)))}{x^2} \, dx\) [189]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {1-\frac {1}{b d n}}{x}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x} \]

[Out]

(-1+1/b/d/n)/x+(1+exp(2*a*d)*(c*x^n)^(2*b*d))/b/d/n/x/(1-exp(2*a*d)*(c*x^n)^(2*b*d))-2*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/x

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5659, 5657, 516, 470, 371} \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x}+\frac {e^{2 a d} \left (c x^n\right )^{2 b d}+1}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {1-\frac {1}{b d n}}{x} \]

[In]

Int[Coth[d*(a + b*Log[c*x^n])]^2/x^2,x]

[Out]

-((1 - 1/(b*d*n))/x) + (1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*n*x*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*Hypergeo
metric2F1[1, -1/2*1/(b*d*n), 1 - 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n*x)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

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] - Dist[(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]

Rule 516

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] + Dist[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] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 5657

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]

Rule 5659

Int[Coth[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \coth ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{n x} \\ & = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \frac {x^{-1-\frac {1}{n}} \left (-1-e^{2 a d} x^{2 b d}\right )^2}{\left (1-e^{2 a d} x^{2 b d}\right )^2} \, dx,x,c x^n\right )}{n x} \\ & = \frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {\left (e^{-2 a d} \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {1}{n}} \left (\frac {2 e^{2 a d} (1+b d n)}{n}+\frac {2 e^{4 a d} (1-b d n) x^{2 b d}}{n}\right )}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{2 b d n x} \\ & = -\frac {1-\frac {1}{b d n}}{x}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {\left (2 \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {1}{n}}}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{b d n^2 x} \\ & = -\frac {1-\frac {1}{b d n}}{x}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \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 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.92 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.18 \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{2 b d n},2-\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(-1+2 b d n) \left (b d n+\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b d n (-1+2 b d n) x} \]

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])]^2/x^2,x]

[Out]

(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*1/(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)*x)

Maple [F]

\[\int \frac {{\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}}{x^{2}}d x\]

[In]

int(coth(d*(a+b*ln(c*x^n)))^2/x^2,x)

[Out]

int(coth(d*(a+b*ln(c*x^n)))^2/x^2,x)

Fricas [F]

\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x^2,x, algorithm="fricas")

[Out]

integral(coth(b*d*log(c*x^n) + a*d)^2/x^2, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\text {Timed out} \]

[In]

integrate(coth(d*(a+b*ln(c*x**n)))**2/x**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x^2,x, algorithm="maxima")

[Out]

-(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n + 2)/(b*c^(2*b*d)*d*n*x*e^(2*b*d*log(x^n) + 2*a*d) - b*d*
n*x) + integrate(1/(b*c^(b*d)*d*n*x^2*e^(b*d*log(x^n) + a*d) + b*d*n*x^2), x) - integrate(1/(b*c^(b*d)*d*n*x^2
*e^(b*d*log(x^n) + a*d) - b*d*n*x^2), x)

Giac [F]

\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x^2,x, algorithm="giac")

[Out]

integrate(coth((b*log(c*x^n) + a)*d)^2/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^2} \,d x \]

[In]

int(coth(d*(a + b*log(c*x^n)))^2/x^2,x)

[Out]

int(coth(d*(a + b*log(c*x^n)))^2/x^2, x)