3.171 \(\int \coth ^p(a+\frac {\log (x)}{4}) \, dx\)

Optimal. Leaf size=108 \[ e^{-4 a} \left (-e^{2 a} \sqrt {x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt {x}\right )^{1-p}-\frac {e^{-4 a} 2^{1-p} p \left (-e^{2 a} \sqrt {x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (e^{2 a} \sqrt {x}+1\right )\right )}{p+1} \]

[Out]

-2^(1-p)*p*hypergeom([p, 1+p],[2+p],1/2+1/2*exp(2*a)*x^(1/2))*(-1-exp(2*a)*x^(1/2))^(1+p)/exp(4*a)/(1+p)+(-1-e
xp(2*a)*x^(1/2))^(1+p)*(1-exp(2*a)*x^(1/2))^(1-p)/exp(4*a)

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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 = {} \[ \int \coth ^p\left (a+\frac {\log (x)}{4}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + Log[x]/4]^p,x]

[Out]

Defer[Int][Coth[(4*a + Log[x])/4]^p, x]

Rubi steps

\begin {align*} \int \coth ^p\left (a+\frac {\log (x)}{4}\right ) \, dx &=\int \coth ^p\left (\frac {1}{4} (4 a+\log (x))\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 125, normalized size = 1.16 \[ \frac {e^{-4 a} \left (e^{2 a} \sqrt {x}+1\right )^{1-p} \left (\frac {e^{2 a} \sqrt {x}+1}{e^{2 a} \sqrt {x}-1}\right )^{p-1} \left ((p-1) \left (e^{2 a} \sqrt {x}+1\right )^{p+1}-2^{p+1} p \, _2F_1\left (1-p,-p;2-p;\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt {x}\right )\right )}{p-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + Log[x]/4]^p,x]

[Out]

((1 + E^(2*a)*Sqrt[x])^(1 - p)*((1 + E^(2*a)*Sqrt[x])/(-1 + E^(2*a)*Sqrt[x]))^(-1 + p)*((-1 + p)*(1 + E^(2*a)*
Sqrt[x])^(1 + p) - 2^(1 + p)*p*Hypergeometric2F1[1 - p, -p, 2 - p, 1/2 - (E^(2*a)*Sqrt[x])/2]))/(E^(4*a)*(-1 +
p))

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + \frac {1}{4} \, \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/4*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 1/4*log(x))^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{4} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/4*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 1/4*log(x))^p, x)

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +\frac {\ln \relax (x )}{4}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+1/4*ln(x))^p,x)

[Out]

int(coth(a+1/4*ln(x))^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{4} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/4*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 1/4*log(x))^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {coth}\left (a+\frac {\ln \relax (x)}{4}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + log(x)/4)^p,x)

[Out]

int(coth(a + log(x)/4)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (a + \frac {\log {\relax (x )}}{4} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/4*ln(x))**p,x)

[Out]

Integral(coth(a + log(x)/4)**p, x)

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