3.172 \(\int \coth ^p(a+\frac{\log (x)}{6}) \, dx\)

Optimal. Leaf size=162 \[ -\frac{e^{-6 a} 2^{-p} \left (2 p^2+1\right ) \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{1}{2} \left (e^{2 a} \sqrt [3]{x}+1\right )\right )}{p+1}+e^{-6 a} p \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p}+e^{-4 a} \sqrt [3]{x} \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p} \]

[Out]

(p*(-1 - E^(2*a)*x^(1/3))^(1 + p)*(1 - E^(2*a)*x^(1/3))^(1 - p))/E^(6*a) + ((-1 - E^(2*a)*x^(1/3))^(1 + p)*(1
- E^(2*a)*x^(1/3))^(1 - p)*x^(1/3))/E^(4*a) - ((1 + 2*p^2)*(-1 - E^(2*a)*x^(1/3))^(1 + p)*Hypergeometric2F1[p,
1 + p, 2 + p, (1 + E^(2*a)*x^(1/3))/2])/(2^p*E^(6*a)*(1 + p))

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Rubi [F]  time = 0.0499177, 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 = {} \[ \int \coth ^p\left (a+\frac{\log (x)}{6}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + Log[x]/6]^p,x]

[Out]

Defer[Int][Coth[(6*a + Log[x])/6]^p, x]

Rubi steps

\begin{align*} \int \coth ^p\left (a+\frac{\log (x)}{6}\right ) \, dx &=\int \coth ^p\left (\frac{1}{6} (6 a+\log (x))\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 0.561147, size = 142, normalized size = 0.88 \[ \frac{e^{-6 a} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p} \left (\frac{e^{2 a} \sqrt [3]{x}+1}{e^{2 a} \sqrt [3]{x}-1}\right )^{p-1} \left ((p-1) \left (e^{2 a} \sqrt [3]{x}+1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+p\right )-2^p \left (2 p^2+1\right ) \, _2F_1\left (1-p,-p;2-p;\frac{1}{2}-\frac{1}{2} e^{2 a} \sqrt [3]{x}\right )\right )}{p-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + Log[x]/6]^p,x]

[Out]

((1 + E^(2*a)*x^(1/3))^(1 - p)*((1 + E^(2*a)*x^(1/3))/(-1 + E^(2*a)*x^(1/3)))^(-1 + p)*((-1 + p)*(1 + E^(2*a)*
x^(1/3))^(1 + p)*(p + E^(2*a)*x^(1/3)) - 2^p*(1 + 2*p^2)*Hypergeometric2F1[1 - p, -p, 2 - p, 1/2 - (E^(2*a)*x^
(1/3))/2]))/(E^(6*a)*(-1 + p))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (a+{\frac{\ln \left ( x \right ) }{6}}\right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+1/6*ln(x))^p,x)

[Out]

int(coth(a+1/6*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}{6} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 1/6*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}{6} \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 1/6*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 )}}{6} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*ln(x))**p,x)

[Out]

Integral(coth(a + log(x)/6)**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (a + \frac{1}{6} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 1/6*log(x))^p, x)