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

Optimal. Leaf size=52 \[ -\frac{e^{-2 a} 2^{-p} \left (-e^{2 a} x-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{1}{2} \left (e^{2 a} x+1\right )\right )}{p+1} \]

[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)))

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Rubi [F]  time = 0.046479, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \coth ^p\left (a+\frac{\log (x)}{2}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + Log[x]/2]^p,x]

[Out]

Defer[Int][Coth[(2*a + Log[x])/2]^p, x]

Rubi steps

\begin{align*} \int \coth ^p\left (a+\frac{\log (x)}{2}\right ) \, dx &=\int \coth ^p\left (\frac{1}{2} (2 a+\log (x))\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 0.391243, size = 83, normalized size = 1.6 \[ -\frac{e^{-2 a} 2^p \left (e^{2 a} x+1\right )^{1-p} \left (\frac{e^{2 a} x+1}{e^{2 a} x-1}\right )^{p-1} \, _2F_1\left (1-p,-p;2-p;\frac{1}{2}-\frac{1}{2} e^{2 a} x\right )}{p-1} \]

Warning: Unable to verify antiderivative.

[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)))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (a+{\frac{\ln \left ( x \right ) }{2}}\right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(a + 1/2*log(x))^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\coth \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(coth(a + 1/2*log(x))^p, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth ^{p}{\left (a + \frac{\log{\left (x \right )}}{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + log(x)/2)**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (a + \frac{1}{2} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(a + 1/2*log(x))^p, x)