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

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*}

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Mathematica [B]  time = 2.24, 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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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (b \log \relax (x) + a\right )^{p}, x\right ) \]

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

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)

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maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}\left (a +b \ln \relax (x )\right )\, dx \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*log(x))^p,x)

[Out]

int(tanh(a + b*log(x))^p, x)

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

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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