\(\int \coth ^p(a+\frac {\log (x)}{2}) \, dx\) [170]

Optimal result

Integrand size = 11, antiderivative size = 52 \[ \int \coth ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=-\frac {2^{-p} e^{-2 a} \left (-1-e^{2 a} x\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1+e^{2 a} x\right )\right )}{1+p} \]

[Out]

-(-1-exp(2*a)*x)^(p+1)*hypergeom([p, p+1],[2+p],1/2+1/2*exp(2*a)*x)/(2^p)/exp(2*a)/(p+1)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5653, 71} \[ \int \coth ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=-\frac {e^{-2 a} 2^{-p} \left (-e^{2 a} x-1\right )^{p+1} \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,\frac {1}{2} \left (e^{2 a} x+1\right )\right )}{p+1} \]

[In]

Int[Coth[a + Log[x]/2]^p,x]

[Out]

-(((-1 - E^(2*a)*x)^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1 + E^(2*a)*x)/2])/(2^p*E^(2*a)*(1 + p)))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 5653

Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(-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, p}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-1-e^{2 a} x\right )^p \left (1-e^{2 a} x\right )^{-p} \, dx \\ & = -\frac {2^{-p} e^{-2 a} \left (-1-e^{2 a} x\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1+e^{2 a} x\right )\right )}{1+p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.60 \[ \int \coth ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=-\frac {2^p e^{-2 a} \left (1+e^{2 a} x\right )^{1-p} \left (\frac {1+e^{2 a} x}{-1+e^{2 a} x}\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,\frac {1}{2}-\frac {1}{2} e^{2 a} x\right )}{-1+p} \]

[In]

Integrate[Coth[a + Log[x]/2]^p,x]

[Out]

-((2^p*(1 + E^(2*a)*x)^(1 - p)*((1 + E^(2*a)*x)/(-1 + E^(2*a)*x))^(-1 + p)*Hypergeometric2F1[1 - p, -p, 2 - p,
1/2 - (E^(2*a)*x)/2])/(E^(2*a)*(-1 + p)))

Maple [F]

\[\int \coth \left (a +\frac {\ln \left (x \right )}{2}\right )^{p}d x\]

[In]

int(coth(a+1/2*ln(x))^p,x)

[Out]

int(coth(a+1/2*ln(x))^p,x)

Fricas [F]

\[ \int \coth ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int { \coth \left (a + \frac {1}{2} \, \log \left (x\right )\right )^{p} \,d x } \]

[In]

integrate(coth(a+1/2*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 1/2*log(x))^p, x)

Sympy [F]

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

[In]

integrate(coth(a+1/2*ln(x))**p,x)

[Out]

Integral(coth(a + log(x)/2)**p, x)

Maxima [F]

\[ \int \coth ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int { \coth \left (a + \frac {1}{2} \, \log \left (x\right )\right )^{p} \,d x } \]

[In]

integrate(coth(a+1/2*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 1/2*log(x))^p, x)

Giac [F]

\[ \int \coth ^p\left (a+\frac {\log (x)}{2}\right ) \, dx=\int { \coth \left (a + \frac {1}{2} \, \log \left (x\right )\right )^{p} \,d x } \]

[In]

integrate(coth(a+1/2*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 1/2*log(x))^p, x)

Mupad [F(-1)]

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

[In]

int(coth(a + log(x)/2)^p,x)

[Out]

int(coth(a + log(x)/2)^p, x)