3.166 \(\int \tanh ^p(a+\frac {\log (x)}{4}) \, dx\)

Optimal. Leaf size=106 \[ e^{-4 a} \left (e^{2 a} \sqrt {x}-1\right )^{p+1} \left (e^{2 a} \sqrt {x}+1\right )^{1-p}-\frac {e^{-4 a} 2^{1-p} p \left (e^{2 a} \sqrt {x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (1-e^{2 a} \sqrt {x}\right )\right )}{p+1} \]

[Out]

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

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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 = {} \[ \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + Log[x]/4]^p,x]

[Out]

Defer[Int][Tanh[(4*a + Log[x])/4]^p, x]

Rubi steps

\begin {align*} \int \tanh ^p\left (a+\frac {\log (x)}{4}\right ) \, dx &=\int \tanh ^p\left (\frac {1}{4} (4 a+\log (x))\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 3.10, size = 121, normalized size = 1.14 \[ \frac {e^{-4 a} \left (e^{2 a} \sqrt {x}-1\right ) \left (\frac {e^{2 a} \sqrt {x}-1}{2 e^{2 a} \sqrt {x}+2}\right )^p \left (2^p (p+1) \left (e^{2 a} \sqrt {x}+1\right )-2 p \left (e^{2 a} \sqrt {x}+1\right )^p \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (1-e^{2 a} \sqrt {x}\right )\right )\right )}{p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + Log[x]/4]^p,x]

[Out]

((-1 + E^(2*a)*Sqrt[x])*((-1 + E^(2*a)*Sqrt[x])/(2 + 2*E^(2*a)*Sqrt[x]))^p*(2^p*(1 + p)*(1 + E^(2*a)*Sqrt[x])
- 2*p*(1 + E^(2*a)*Sqrt[x])^p*Hypergeometric2F1[p, 1 + p, 2 + p, (1 - E^(2*a)*Sqrt[x])/2]))/(E^(4*a)*(1 + p))

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (a + \frac {1}{4} \, \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/4*log(x))^p,x, algorithm="fricas")

[Out]

integral(tanh(a + 1/4*log(x))^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left (a + \frac {1}{4} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/4*log(x))^p,x, algorithm="giac")

[Out]

integrate(tanh(a + 1/4*log(x))^p, x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}\left (a +\frac {\ln \relax (x )}{4}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a+1/4*ln(x))^p,x)

[Out]

int(tanh(a+1/4*ln(x))^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left (a + \frac {1}{4} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/4*log(x))^p,x, algorithm="maxima")

[Out]

integrate(tanh(a + 1/4*log(x))^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tanh}\left (a+\frac {\ln \relax (x)}{4}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + log(x)/4)^p,x)

[Out]

int(tanh(a + log(x)/4)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}{\left (a + \frac {\log {\relax (x )}}{4} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/4*ln(x))**p,x)

[Out]

Integral(tanh(a + log(x)/4)**p, x)

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