3.168 \(\int \tanh ^p(a+\frac {\log (x)}{8}) \, dx\)

Optimal. Leaf size=190 \[ -\frac {e^{-8 a} 2^{2-p} p \left (p^2+2\right ) \left (e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (1-e^{2 a} \sqrt [4]{x}\right )\right )}{3 (p+1)}+\frac {1}{3} e^{-12 a} \left (e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{4 a} \left (2 p^2+3\right )-2 e^{6 a} p \sqrt [4]{x}\right ) \left (e^{2 a} \sqrt [4]{x}+1\right )^{1-p}+e^{-4 a} \sqrt {x} \left (e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{2 a} \sqrt [4]{x}+1\right )^{1-p} \]

[Out]

1/3*(-1+exp(2*a)*x^(1/4))^(1+p)*(1+exp(2*a)*x^(1/4))^(1-p)*(exp(4*a)*(2*p^2+3)-2*exp(6*a)*p*x^(1/4))/exp(12*a)
-1/3*2^(2-p)*p*(p^2+2)*(-1+exp(2*a)*x^(1/4))^(1+p)*hypergeom([p, 1+p],[2+p],1/2-1/2*exp(2*a)*x^(1/4))/exp(8*a)
/(1+p)+(-1+exp(2*a)*x^(1/4))^(1+p)*(1+exp(2*a)*x^(1/4))^(1-p)*x^(1/2)/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 = {} \[ \int \tanh ^p\left (a+\frac {\log (x)}{8}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + Log[x]/8]^p,x]

[Out]

Defer[Int][Tanh[(8*a + Log[x])/8]^p, x]

Rubi steps

\begin {align*} \int \tanh ^p\left (a+\frac {\log (x)}{8}\right ) \, dx &=\int \tanh ^p\left (\frac {1}{8} (8 a+\log (x))\right ) \, dx\\ \end {align*}

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Mathematica [C]  time = 3.41, size = 177, normalized size = 0.93 \[ \frac {5 x \left (\frac {e^{2 a} \sqrt [4]{x}-1}{e^{2 a} \sqrt [4]{x}+1}\right )^p F_1\left (4;-p,p;5;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )}{5 F_1\left (4;-p,p;5;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )-e^{2 a} p \sqrt [4]{x} \left (F_1\left (5;1-p,p;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )+F_1\left (5;-p,p+1;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[a + Log[x]/8]^p,x]

[Out]

(5*((-1 + E^(2*a)*x^(1/4))/(1 + E^(2*a)*x^(1/4)))^p*x*AppellF1[4, -p, p, 5, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4)
)])/(5*AppellF1[4, -p, p, 5, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4))] - E^(2*a)*p*x^(1/4)*(AppellF1[5, 1 - p, p, 6
, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4))] + AppellF1[5, -p, 1 + p, 6, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4))]))

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fricas [F]  time = 2.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (a + \frac {1}{8} \, \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/8*log(x))^p,x, algorithm="fricas")

[Out]

integral(tanh(a + 1/8*log(x))^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left (a + \frac {1}{8} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/8*log(x))^p,x, algorithm="giac")

[Out]

integrate(tanh(a + 1/8*log(x))^p, x)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}\left (a +\frac {\ln \relax (x )}{8}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a+1/8*ln(x))^p,x)

[Out]

int(tanh(a+1/8*ln(x))^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left (a + \frac {1}{8} \, \log \relax (x)\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/8*log(x))^p,x, algorithm="maxima")

[Out]

integrate(tanh(a + 1/8*log(x))^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tanh}\left (a+\frac {\ln \relax (x)}{8}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + log(x)/8)^p,x)

[Out]

int(tanh(a + log(x)/8)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}{\left (a + \frac {\log {\relax (x )}}{8} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/8*ln(x))**p,x)

[Out]

Integral(tanh(a + log(x)/8)**p, x)

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