3.174 \(\int \coth ^p(a+\log (x)) \, dx\)

Optimal. Leaf size=61 \[ x \left (-e^{2 a} x^2-1\right )^p \left (e^{2 a} x^2+1\right )^{-p} F_1\left (\frac{1}{2};p,-p;\frac{3}{2};e^{2 a} x^2,-e^{2 a} x^2\right ) \]

[Out]

(x*(-1 - E^(2*a)*x^2)^p*AppellF1[1/2, p, -p, 3/2, E^(2*a)*x^2, -(E^(2*a)*x^2)])/(1 + E^(2*a)*x^2)^p

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Rubi [F]  time = 0.0153368, 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 = {} \[ \int \coth ^p(a+\log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + Log[x]]^p,x]

[Out]

Defer[Int][Coth[a + Log[x]]^p, x]

Rubi steps

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

Mathematica [B]  time = 1.54812, size = 171, normalized size = 2.8 \[ \frac{3 x \left (\frac{e^{2 a} x^2+1}{e^{2 a} x^2-1}\right )^p F_1\left (\frac{1}{2};p,-p;\frac{3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )}{2 e^{2 a} p x^2 \left (F_1\left (\frac{3}{2};p,1-p;\frac{5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )+F_1\left (\frac{3}{2};p+1,-p;\frac{5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )\right )+3 F_1\left (\frac{1}{2};p,-p;\frac{3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + Log[x]]^p,x]

[Out]

(3*x*((1 + E^(2*a)*x^2)/(-1 + E^(2*a)*x^2))^p*AppellF1[1/2, p, -p, 3/2, E^(2*a)*x^2, -(E^(2*a)*x^2)])/(3*Appel
lF1[1/2, p, -p, 3/2, E^(2*a)*x^2, -(E^(2*a)*x^2)] + 2*E^(2*a)*p*x^2*(AppellF1[3/2, p, 1 - p, 5/2, E^(2*a)*x^2,
-(E^(2*a)*x^2)] + AppellF1[3/2, 1 + p, -p, 5/2, E^(2*a)*x^2, -(E^(2*a)*x^2)]))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (a+\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+ln(x))^p,x)

[Out]

int(coth(a+ln(x))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+log(x))^p,x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + log(x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+ln(x))**p,x)

[Out]

Integral(coth(a + log(x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + log(x))^p, x)