3.172 \(\int \coth ^p(a+\frac {\log (x)}{6}) \, dx\)

Optimal. Leaf size=162 \[ -\frac {e^{-6 a} 2^{-p} \left (2 p^2+1\right ) \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (e^{2 a} \sqrt [3]{x}+1\right )\right )}{p+1}+e^{-6 a} p \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p}+e^{-4 a} \sqrt [3]{x} \left (-e^{2 a} \sqrt [3]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [3]{x}\right )^{1-p} \]

[Out]

p*(-1-exp(2*a)*x^(1/3))^(1+p)*(1-exp(2*a)*x^(1/3))^(1-p)/exp(6*a)+(-1-exp(2*a)*x^(1/3))^(1+p)*(1-exp(2*a)*x^(1
/3))^(1-p)*x^(1/3)/exp(4*a)-(2*p^2+1)*(-1-exp(2*a)*x^(1/3))^(1+p)*hypergeom([p, 1+p],[2+p],1/2+1/2*exp(2*a)*x^
(1/3))/(2^p)/exp(6*a)/(1+p)

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Rubi [F]  time = 0.05, 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 = {} \[ \int \coth ^p\left (a+\frac {\log (x)}{6}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + Log[x]/6]^p,x]

[Out]

Defer[Int][Coth[(6*a + Log[x])/6]^p, x]

Rubi steps

\begin {align*} \int \coth ^p\left (a+\frac {\log (x)}{6}\right ) \, dx &=\int \coth ^p\left (\frac {1}{6} (6 a+\log (x))\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 142, normalized size = 0.88 \[ \frac {e^{-6 a} \left (e^{2 a} \sqrt [3]{x}+1\right )^{1-p} \left (\frac {e^{2 a} \sqrt [3]{x}+1}{e^{2 a} \sqrt [3]{x}-1}\right )^{p-1} \left ((p-1) \left (e^{2 a} \sqrt [3]{x}+1\right )^{p+1} \left (e^{2 a} \sqrt [3]{x}+p\right )-2^p \left (2 p^2+1\right ) \, _2F_1\left (1-p,-p;2-p;\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [3]{x}\right )\right )}{p-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + Log[x]/6]^p,x]

[Out]

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

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fricas [F]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (a + \frac {1}{6} \, \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 1/6*log(x))^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{6} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 1/6*log(x))^p, x)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (a +\frac {\ln \relax (x )}{6}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+1/6*ln(x))^p,x)

[Out]

int(coth(a+1/6*ln(x))^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (a + \frac {1}{6} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 1/6*log(x))^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {coth}\left (a+\frac {\ln \relax (x)}{6}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + log(x)/6)^p,x)

[Out]

int(coth(a + log(x)/6)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (a + \frac {\log {\relax (x )}}{6} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/6*ln(x))**p,x)

[Out]

Integral(coth(a + log(x)/6)**p, x)

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