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

Optimal. Leaf size=79 \[ x \left (1-e^{2 a} x^{2 b}\right )^{-p} \left (e^{2 a} x^{2 b}-1\right )^p F_1\left (\frac{1}{2 b};-p,p;\frac{1}{2} \left (2+\frac{1}{b}\right );e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right ) \]

[Out]

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

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Rubi [F]  time = 0.0229634, 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 = {} \[ \int \tanh ^p(a+b \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + b*Log[x]]^p,x]

[Out]

Defer[Int][Tanh[a + b*Log[x]]^p, x]

Rubi steps

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

Mathematica [B]  time = 1.90654, size = 259, normalized size = 3.28 \[ \frac{(2 b+1) x \left (\frac{e^{2 a} x^{2 b}-1}{e^{2 a} x^{2 b}+1}\right )^p F_1\left (\frac{1}{2 b};-p,p;1+\frac{1}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{-2 e^{2 a} b p x^{2 b} F_1\left (1+\frac{1}{2 b};1-p,p;2+\frac{1}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )-2 e^{2 a} b p x^{2 b} F_1\left (1+\frac{1}{2 b};-p,p+1;2+\frac{1}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )+(2 b+1) F_1\left (\frac{1}{2 b};-p,p;1+\frac{1}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[a + b*Log[x]]^p,x]

[Out]

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

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \begin{align*} \int \left ( \tanh \left ( a+b\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(b*log(x) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(tanh(b*log(x) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(a + b*log(x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(b*log(x) + a)^p, x)