3.169 \(\int \tanh ^p(a+\log (x)) \, dx\)

Optimal. Leaf size=61 \[ x \left (1-e^{2 a} x^2\right )^{-p} \left (e^{2 a} x^2-1\right )^p F_1\left (\frac {1}{2};-p,p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right ) \]

[Out]

x*(-1+exp(2*a)*x^2)^p*AppellF1(1/2,-p,p,3/2,exp(2*a)*x^2,-exp(2*a)*x^2)/((1-exp(2*a)*x^2)^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 = {} \[ \int \tanh ^p(a+\log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + Log[x]]^p,x]

[Out]

Defer[Int][Tanh[a + Log[x]]^p, x]

Rubi steps

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

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Mathematica [B]  time = 1.74, size = 171, normalized size = 2.80 \[ \frac {3 x \left (\frac {e^{2 a} x^2-1}{e^{2 a} x^2+1}\right )^p F_1\left (\frac {1}{2};-p,p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )}{3 F_1\left (\frac {1}{2};-p,p;\frac {3}{2};e^{2 a} x^2,-e^{2 a} x^2\right )-2 e^{2 a} p x^2 \left (F_1\left (\frac {3}{2};1-p,p;\frac {5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )+F_1\left (\frac {3}{2};-p,p+1;\frac {5}{2};e^{2 a} x^2,-e^{2 a} x^2\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[a + Log[x]]^p,x]

[Out]

(3*x*((-1 + E^(2*a)*x^2)/(1 + E^(2*a)*x^2))^p*AppellF1[1/2, -p, p, 3/2, E^(2*a)*x^2, -(E^(2*a)*x^2)])/(3*Appel
lF1[1/2, -p, p, 3/2, E^(2*a)*x^2, -(E^(2*a)*x^2)] - 2*E^(2*a)*p*x^2*(AppellF1[3/2, 1 - p, p, 5/2, E^(2*a)*x^2,
-(E^(2*a)*x^2)] + AppellF1[3/2, -p, 1 + p, 5/2, E^(2*a)*x^2, -(E^(2*a)*x^2)]))

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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (a + \log \relax (x)\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+log(x))^p,x, algorithm="fricas")

[Out]

integral(tanh(a + log(x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+log(x))^p,x, algorithm="giac")

[Out]

integrate(tanh(a + log(x))^p, x)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}\left (a +\ln \relax (x )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a+ln(x))^p,x)

[Out]

int(tanh(a+ln(x))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+log(x))^p,x, algorithm="maxima")

[Out]

integrate(tanh(a + log(x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + log(x))^p,x)

[Out]

int(tanh(a + log(x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+ln(x))**p,x)

[Out]

Integral(tanh(a + log(x))**p, x)

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