3.176 \(\int \coth ^p(a+3 \log (x)) \, dx\)

Optimal. Leaf size=61 \[ x \left (-e^{2 a} x^6-1\right )^p \left (e^{2 a} x^6+1\right )^{-p} F_1\left (\frac {1}{6};p,-p;\frac {7}{6};e^{2 a} x^6,-e^{2 a} x^6\right ) \]

[Out]

x*(-1-exp(2*a)*x^6)^p*AppellF1(1/6,p,-p,7/6,exp(2*a)*x^6,-exp(2*a)*x^6)/((1+exp(2*a)*x^6)^p)

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Rubi [F]  time = 0.02, 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 = {} \[ \int \coth ^p(a+3 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 3*Log[x]]^p,x]

[Out]

Defer[Int][Coth[a + 3*Log[x]]^p, x]

Rubi steps

\begin {align*} \int \coth ^p(a+3 \log (x)) \, dx &=\int \coth ^p(a+3 \log (x)) \, dx\\ \end {align*}

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Mathematica [B]  time = 2.01, size = 171, normalized size = 2.80 \[ \frac {7 x \left (\frac {e^{2 a} x^6+1}{e^{2 a} x^6-1}\right )^p F_1\left (\frac {1}{6};p,-p;\frac {7}{6};e^{2 a} x^6,-e^{2 a} x^6\right )}{6 e^{2 a} p x^6 \left (F_1\left (\frac {7}{6};p,1-p;\frac {13}{6};e^{2 a} x^6,-e^{2 a} x^6\right )+F_1\left (\frac {7}{6};p+1,-p;\frac {13}{6};e^{2 a} x^6,-e^{2 a} x^6\right )\right )+7 F_1\left (\frac {1}{6};p,-p;\frac {7}{6};e^{2 a} x^6,-e^{2 a} x^6\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + 3*Log[x]]^p,x]

[Out]

(7*x*((1 + E^(2*a)*x^6)/(-1 + E^(2*a)*x^6))^p*AppellF1[1/6, p, -p, 7/6, E^(2*a)*x^6, -(E^(2*a)*x^6)])/(7*Appel
lF1[1/6, p, -p, 7/6, E^(2*a)*x^6, -(E^(2*a)*x^6)] + 6*E^(2*a)*p*x^6*(AppellF1[7/6, p, 1 - p, 13/6, E^(2*a)*x^6
, -(E^(2*a)*x^6)] + AppellF1[7/6, 1 + p, -p, 13/6, E^(2*a)*x^6, -(E^(2*a)*x^6)]))

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + 3 \, \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+3*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 3*log(x))^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + 3 \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+3*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 3*log(x))^p, x)

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maple [F]  time = 0.19, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +3 \ln \relax (x )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+3*ln(x))^p,x)

[Out]

int(coth(a+3*ln(x))^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + 3 \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+3*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 3*log(x))^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {coth}\left (a+3\,\ln \relax (x)\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 3*log(x))^p,x)

[Out]

int(coth(a + 3*log(x))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (a + 3 \log {\relax (x )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+3*ln(x))**p,x)

[Out]

Integral(coth(a + 3*log(x))**p, x)

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