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

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Rubi [F]  time = 0.01, 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 = {} \[ \int \coth ^p(a+2 \log (x)) \, dx \]

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*}

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Mathematica [B]  time = 1.95, size = 171, normalized size = 2.80 \[ \frac {5 x \left (\frac {e^{2 a} x^4+1}{e^{2 a} x^4-1}\right )^p F_1\left (\frac {1}{4};p,-p;\frac {5}{4};e^{2 a} x^4,-e^{2 a} x^4\right )}{4 e^{2 a} p x^4 \left (F_1\left (\frac {5}{4};p,1-p;\frac {9}{4};e^{2 a} x^4,-e^{2 a} x^4\right )+F_1\left (\frac {5}{4};p+1,-p;\frac {9}{4};e^{2 a} x^4,-e^{2 a} x^4\right )\right )+5 F_1\left (\frac {1}{4};p,-p;\frac {5}{4};e^{2 a} x^4,-e^{2 a} x^4\right )} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + 2 \, \log \relax (x)\right )^{p}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + 2 \, \log \relax (x)\right )^{p}\,{d x} \]

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)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +2 \ln \relax (x )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + 2 \, \log \relax (x)\right )^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {coth}\left (a+2\,\ln \relax (x)\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))^p,x)

[Out]

int(coth(a + 2*log(x))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]

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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