3.175 \(\int \coth ^p(a+2 \log (x)) \, dx\)

Optimal. Leaf size=61 \[ x \left (-e^{2 a} x^4-1\right )^p \left (e^{2 a} x^4+1\right )^{-p} F_1\left (\frac{1}{4};p,-p;\frac{5}{4};e^{2 a} x^4,-e^{2 a} x^4\right ) \]

[Out]

(x*(-1 - E^(2*a)*x^4)^p*AppellF1[1/4, p, -p, 5/4, E^(2*a)*x^4, -(E^(2*a)*x^4)])/(1 + E^(2*a)*x^4)^p

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Rubi [F]  time = 0.0105149, 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 = {} \[ \int \coth ^p(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]^p,x]

[Out]

Defer[Int][Coth[a + 2*Log[x]]^p, x]

Rubi steps

\begin{align*} \int \coth ^p(a+2 \log (x)) \, dx &=\int \coth ^p(a+2 \log (x)) \, dx\\ \end{align*}

Mathematica [B]  time = 1.7343, size = 171, normalized size = 2.8 \[ \frac{5 x \left (\frac{e^{2 a} x^4+1}{e^{2 a} x^4-1}\right )^p F_1\left (\frac{1}{4};p,-p;\frac{5}{4};e^{2 a} x^4,-e^{2 a} x^4\right )}{4 e^{2 a} p x^4 \left (F_1\left (\frac{5}{4};p,1-p;\frac{9}{4};e^{2 a} x^4,-e^{2 a} x^4\right )+F_1\left (\frac{5}{4};p+1,-p;\frac{9}{4};e^{2 a} x^4,-e^{2 a} x^4\right )\right )+5 F_1\left (\frac{1}{4};p,-p;\frac{5}{4};e^{2 a} x^4,-e^{2 a} x^4\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + 2*Log[x]]^p,x]

[Out]

(5*x*((1 + E^(2*a)*x^4)/(-1 + E^(2*a)*x^4))^p*AppellF1[1/4, p, -p, 5/4, E^(2*a)*x^4, -(E^(2*a)*x^4)])/(5*Appel
lF1[1/4, p, -p, 5/4, E^(2*a)*x^4, -(E^(2*a)*x^4)] + 4*E^(2*a)*p*x^4*(AppellF1[5/4, p, 1 - p, 9/4, E^(2*a)*x^4,
-(E^(2*a)*x^4)] + AppellF1[5/4, 1 + p, -p, 9/4, E^(2*a)*x^4, -(E^(2*a)*x^4)]))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+2*ln(x))^p,x)

[Out]

int(coth(a+2*ln(x))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (a + 2 \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 2*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 + 2 \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 2*log(x))^p, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth ^{p}{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*ln(x))**p,x)

[Out]

Integral(coth(a + 2*log(x))**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth \left (a + 2 \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 2*log(x))^p, x)