3.173 \(\int \coth ^p(a+\frac{\log (x)}{8}) \, dx\)

Optimal. Leaf size=194 \[ -\frac{e^{-8 a} 2^{2-p} p \left (p^2+2\right ) \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{1}{2} \left (e^{2 a} \sqrt [4]{x}+1\right )\right )}{3 (p+1)}+\frac{1}{3} e^{-12 a} \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{4 a} \left (2 p^2+3\right )+2 e^{6 a} p \sqrt [4]{x}\right ) \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p}+e^{-4 a} \sqrt{x} \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \]

[Out]

((-1 - E^(2*a)*x^(1/4))^(1 + p)*(1 - E^(2*a)*x^(1/4))^(1 - p)*(E^(4*a)*(3 + 2*p^2) + 2*E^(6*a)*p*x^(1/4)))/(3*
E^(12*a)) + ((-1 - E^(2*a)*x^(1/4))^(1 + p)*(1 - E^(2*a)*x^(1/4))^(1 - p)*Sqrt[x])/E^(4*a) - (2^(2 - p)*p*(2 +
p^2)*(-1 - E^(2*a)*x^(1/4))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1 + E^(2*a)*x^(1/4))/2])/(3*E^(8*a)*(
1 + p))

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Rubi [F]  time = 0.0507109, 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 = {} \[ \int \coth ^p\left (a+\frac{\log (x)}{8}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + Log[x]/8]^p,x]

[Out]

Defer[Int][Coth[(8*a + Log[x])/8]^p, x]

Rubi steps

\begin{align*} \int \coth ^p\left (a+\frac{\log (x)}{8}\right ) \, dx &=\int \coth ^p\left (\frac{1}{8} (8 a+\log (x))\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 0.906553, size = 223, normalized size = 1.15 \[ \frac{e^{-8 a} \left (e^{2 a} \sqrt [4]{x}+1\right )^{1-p} \left (\frac{e^{2 a} \sqrt [4]{x}+1}{e^{2 a} \sqrt [4]{x}-1}\right )^{p-1} \left (-2^{p+3} p \, _2F_1\left (-p-2,1-p;2-p;\frac{1}{2}-\frac{1}{2} e^{2 a} \sqrt [4]{x}\right )+2^{p+2} (2 p-1) \, _2F_1\left (-p-1,1-p;2-p;\frac{1}{2}-\frac{1}{2} e^{2 a} \sqrt [4]{x}\right )+(p-1) \left (e^{4 a} \sqrt{x} \left (e^{2 a} \sqrt [4]{x}+1\right )^{p+1}-2^{p+1} \, _2F_1\left (1-p,-p;2-p;\frac{1}{2}-\frac{1}{2} e^{2 a} \sqrt [4]{x}\right )\right )\right )}{p-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + Log[x]/8]^p,x]

[Out]

((1 + E^(2*a)*x^(1/4))^(1 - p)*((1 + E^(2*a)*x^(1/4))/(-1 + E^(2*a)*x^(1/4)))^(-1 + p)*(-(2^(3 + p)*p*Hypergeo
metric2F1[-2 - p, 1 - p, 2 - p, 1/2 - (E^(2*a)*x^(1/4))/2]) + 2^(2 + p)*(-1 + 2*p)*Hypergeometric2F1[-1 - p, 1
- p, 2 - p, 1/2 - (E^(2*a)*x^(1/4))/2] + (-1 + p)*(E^(4*a)*(1 + E^(2*a)*x^(1/4))^(1 + p)*Sqrt[x] - 2^(1 + p)*
Hypergeometric2F1[1 - p, -p, 2 - p, 1/2 - (E^(2*a)*x^(1/4))/2])))/(E^(8*a)*(-1 + p))

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (a+{\frac{\ln \left ( x \right ) }{8}}\right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+1/8*ln(x))^p,x)

[Out]

int(coth(a+1/8*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}{8} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/8*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 1/8*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}{8} \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/8*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 1/8*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 )}}{8} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/8*ln(x))**p,x)

[Out]

Integral(coth(a + log(x)/8)**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (a + \frac{1}{8} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/8*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 1/8*log(x))^p, x)