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]

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

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Rubi [F]  time = 0.0527003, 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 = {} \[ \int \tanh ^p\left (a+\frac{\log (x)}{4}\right ) \, dx \]

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*}

Mathematica [A]  time = 2.90407, 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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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int \left ( \tanh \left ( a+{\frac{\ln \left ( x \right ) }{4}} \right ) \right ) ^{p}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (a + \frac{1}{4} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\tanh \left (a + \frac{1}{4} \, \log \left (x\right )\right )^{p}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh ^{p}{\left (a + \frac{\log{\left (x \right )}}{4} \right )}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (a + \frac{1}{4} \, \log \left (x\right )\right )^{p}\,{d x} \end{align*}

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)