3.2.70 \(\int \tanh ^p(a+2 \log (x)) \, dx\) [170]

Optimal. Leaf size=61 \[ x \left (1-e^{2 a} x^4\right )^{-p} \left (-1+e^{2 a} x^4\right )^p F_1\left (\frac {1}{4};-p,p;\frac {5}{4};e^{2 a} x^4,-e^{2 a} x^4\right ) \]

[Out]

x*(-1+exp(2*a)*x^4)^p*AppellF1(1/4,-p,p,5/4,exp(2*a)*x^4,-exp(2*a)*x^4)/((1-exp(2*a)*x^4)^p)

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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5652, 441, 440} \begin {gather*} x \left (1-e^{2 a} x^4\right )^{-p} \left (e^{2 a} x^4-1\right )^p F_1\left (\frac {1}{4};-p,p;\frac {5}{4};e^{2 a} x^4,-e^{2 a} x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[a + 2*Log[x]]^p,x]

[Out]

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

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 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(a+2 \log (x)) \, dx &=\int \tanh ^p(a+2 \log (x)) \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(61)=122\).
time = 1.32, size = 171, normalized size = 2.80 \begin {gather*} \frac {5 x \left (\frac {-1+e^{2 a} x^4}{1+e^{2 a} x^4}\right )^p F_1\left (\frac {1}{4};-p,p;\frac {5}{4};e^{2 a} x^4,-e^{2 a} x^4\right )}{5 F_1\left (\frac {1}{4};-p,p;\frac {5}{4};e^{2 a} x^4,-e^{2 a} x^4\right )-4 e^{2 a} p x^4 \left (F_1\left (\frac {5}{4};1-p,p;\frac {9}{4};e^{2 a} x^4,-e^{2 a} x^4\right )+F_1\left (\frac {5}{4};-p,1+p;\frac {9}{4};e^{2 a} x^4,-e^{2 a} x^4\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[a + 2*Log[x]]^p,x]

[Out]

(5*x*((-1 + E^(2*a)*x^4)/(1 + E^(2*a)*x^4))^p*AppellF1[1/4, -p, p, 5/4, E^(2*a)*x^4, -(E^(2*a)*x^4)])/(5*Appel
lF1[1/4, -p, p, 5/4, E^(2*a)*x^4, -(E^(2*a)*x^4)] - 4*E^(2*a)*p*x^4*(AppellF1[5/4, 1 - p, p, 9/4, E^(2*a)*x^4,
-(E^(2*a)*x^4)] + AppellF1[5/4, -p, 1 + p, 9/4, E^(2*a)*x^4, -(E^(2*a)*x^4)]))

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Maple [F]
time = 0.62, size = 0, normalized size = 0.00 \[\int \tanh ^{p}\left (a +2 \ln \left (x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(tanh(a + 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 + 2 \log {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(tanh(a + 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+2\,\ln \left (x\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + 2*log(x))^p,x)

[Out]

int(tanh(a + 2*log(x))^p, x)

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