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

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Rubi [F]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.43, size = 83, normalized size = 1.60 \[ -\frac {e^{-2 a} 2^p \left (e^{2 a} x+1\right )^{1-p} \left (\frac {e^{2 a} x+1}{e^{2 a} x-1}\right )^{p-1} \, _2F_1\left (1-p,-p;2-p;\frac {1}{2}-\frac {1}{2} e^{2 a} x\right )}{p-1} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + \frac {1}{2} \, \log \relax (x)\right )^{p}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{2} \, \log \relax (x)\right )^{p}\,{d x} \]

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)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +\frac {\ln \relax (x )}{2}\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{2} \, \log \relax (x)\right )^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {coth}\left (a+\frac {\ln \relax (x)}{2}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + log(x)/2)^p,x)

[Out]

int(coth(a + log(x)/2)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (a + \frac {\log {\relax (x )}}{2} \right )}\, dx \]

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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