3.2.68 \(\int \coth ^p(a+b \log (x)) \, dx\) [168]

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

[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 [A]
time = 0.03, antiderivative size = 79, 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 = {5653, 441, 440} \begin {gather*} x \left (-e^{2 a} x^{2 b}-1\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*Log[x]]^p,x]

[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

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 5653

Int[Coth[((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 \coth ^p(a+b \log (x)) \, dx &=\int \coth ^p(a+b \log (x)) \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(79)=158\).
time = 1.36, size = 259, normalized size = 3.28 \begin {gather*} \frac {(1+2 b) x \left (\frac {1+e^{2 a} x^{2 b}}{-1+e^{2 a} x^{2 b}}\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 b e^{2 a} p x^{2 b} F_1\left (1+\frac {1}{2 b};p,1-p;2+\frac {1}{2 b};e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )+2 b e^{2 a} 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 )+(1+2 b) 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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[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), p, 1 - 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), 1 + p, -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 = 0.85, size = 0, normalized size = 0.00 \[\int \coth ^{p}\left (a +b \ln \left (x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(coth(a+b*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(coth(a+b*log(x))^p,x, algorithm="maxima")

[Out]

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

[Out]

integral(coth(b*log(x) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + b*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(coth(a+b*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(b*log(x) + a)^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (a+b\,\ln \left (x\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*log(x))^p,x)

[Out]

int(coth(a + b*log(x))^p, x)

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