3.2.65 \(\int \tanh ^p(a+\frac {\log (x)}{2}) \, dx\) [165]

Optimal. Leaf size=51 \[ \frac {2^{-p} e^{-2 a} \left (-1+e^{2 a} x\right )^{1+p} \, _2F_1\left (p,1+p;2+p;\frac {1}{2} \left (1-e^{2 a} x\right )\right )}{1+p} \]

[Out]

(-1+exp(2*a)*x)^(1+p)*hypergeom([p, 1+p],[2+p],1/2-1/2*exp(2*a)*x)/(2^p)/exp(2*a)/(1+p)

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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5652, 71} \begin {gather*} \frac {e^{-2 a} 2^{-p} \left (e^{2 a} x-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (1-e^{2 a} x\right )\right )}{p+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[a + Log[x]/2]^p,x]

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 5652

Int[Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(-1 + E^(2*a*d)*x^(2*b*d))^p/(1 + E^(2*a*d)*x^
(2*b*d))^p, x] /; FreeQ[{a, b, d, p}, x]

Rubi steps

\begin {align*} \int \tanh ^p\left (a+\frac {\log (x)}{2}\right ) \, dx &=\int \tanh ^p\left (\frac {1}{2} (2 a+\log (x))\right ) \, dx\\ \end {align*}

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Mathematica [A]
time = 2.08, size = 76, normalized size = 1.49 \begin {gather*} \frac {2^{-p} e^{-2 a} \left (\frac {-1+e^{2 a} x}{1+e^{2 a} x}\right )^{1+p} \left (1+e^{2 a} x\right )^{1+p} \, _2F_1\left (p,1+p;2+p;\frac {1}{2} \left (1-e^{2 a} x\right )\right )}{1+p} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + Log[x]/2]^p,x]

[Out]

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

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Maple [F]
time = 0.61, size = 0, normalized size = 0.00 \[\int \tanh ^{p}\left (a +\frac {\ln \left (x \right )}{2}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a+1/2*ln(x))^p,x)

[Out]

int(tanh(a+1/2*ln(x))^p,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/2*log(x))^p,x, algorithm="maxima")

[Out]

integrate(tanh(a + 1/2*log(x))^p, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/2*log(x))^p,x, algorithm="fricas")

[Out]

integral(tanh(a + 1/2*log(x))^p, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \tanh ^{p}{\left (a + \frac {\log {\left (x \right )}}{2} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/2*ln(x))**p,x)

[Out]

Integral(tanh(a + log(x)/2)**p, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+1/2*log(x))^p,x, algorithm="giac")

[Out]

integrate(tanh(a + 1/2*log(x))^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {tanh}\left (a+\frac {\ln \left (x\right )}{2}\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + log(x)/2)^p,x)

[Out]

int(tanh(a + log(x)/2)^p, x)

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