3.2.73 \(\int \coth ^p(a+\frac {\log (x)}{8}) \, dx\) [173]

Optimal. Leaf size=194 \[ \frac {1}{3} e^{-12 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \left (e^{4 a} \left (3+2 p^2\right )+2 e^{6 a} p \sqrt [4]{x}\right )+e^{-4 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \sqrt {x}-\frac {2^{2-p} e^{-8 a} p \left (2+p^2\right ) \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \, _2F_1\left (p,1+p;2+p;\frac {1}{2} \left (1+e^{2 a} \sqrt [4]{x}\right )\right )}{3 (1+p)} \]

[Out]

1/3*(-1-exp(2*a)*x^(1/4))^(1+p)*(1-exp(2*a)*x^(1/4))^(1-p)*(exp(4*a)*(2*p^2+3)+2*exp(6*a)*p*x^(1/4))/exp(12*a)
-1/3*2^(2-p)*p*(p^2+2)*(-1-exp(2*a)*x^(1/4))^(1+p)*hypergeom([p, 1+p],[2+p],1/2+1/2*exp(2*a)*x^(1/4))/exp(8*a)
/(1+p)+(-1-exp(2*a)*x^(1/4))^(1+p)*(1-exp(2*a)*x^(1/4))^(1-p)*x^(1/2)/exp(4*a)

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Rubi [A]
time = 0.11, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5653, 383, 102, 152, 71} \begin {gather*} -\frac {e^{-8 a} 2^{2-p} p \left (p^2+2\right ) \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \, _2F_1\left (p,p+1;p+2;\frac {1}{2} \left (e^{2 a} \sqrt [4]{x}+1\right )\right )}{3 (p+1)}+\frac {1}{3} e^{-12 a} \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{4 a} \left (2 p^2+3\right )+2 e^{6 a} p \sqrt [4]{x}\right ) \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p}+e^{-4 a} \sqrt {x} \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[a + Log[x]/8]^p,x]

[Out]

((-1 - E^(2*a)*x^(1/4))^(1 + p)*(1 - E^(2*a)*x^(1/4))^(1 - p)*(E^(4*a)*(3 + 2*p^2) + 2*E^(6*a)*p*x^(1/4)))/(3*
E^(12*a)) + ((-1 - E^(2*a)*x^(1/4))^(1 + p)*(1 - E^(2*a)*x^(1/4))^(1 - p)*Sqrt[x])/E^(4*a) - (2^(2 - p)*p*(2 +
p^2)*(-1 - E^(2*a)*x^(1/4))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1 + E^(2*a)*x^(1/4))/2])/(3*E^(8*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 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

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\left (a+\frac {\log (x)}{8}\right ) \, dx &=\int \coth ^p\left (\frac {1}{8} (8 a+\log (x))\right ) \, dx\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 2.04, size = 176, normalized size = 0.91 \begin {gather*} \frac {5 \left (\frac {1+e^{2 a} \sqrt [4]{x}}{-1+e^{2 a} \sqrt [4]{x}}\right )^p x F_1\left (4;p,-p;5;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )}{5 F_1\left (4;p,-p;5;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )+e^{2 a} p \sqrt [4]{x} \left (F_1\left (5;p,1-p;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )+F_1\left (5;1+p,-p;6;e^{2 a} \sqrt [4]{x},-e^{2 a} \sqrt [4]{x}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + Log[x]/8]^p,x]

[Out]

(5*((1 + E^(2*a)*x^(1/4))/(-1 + E^(2*a)*x^(1/4)))^p*x*AppellF1[4, p, -p, 5, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4)
)])/(5*AppellF1[4, p, -p, 5, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4))] + E^(2*a)*p*x^(1/4)*(AppellF1[5, p, 1 - p, 6
, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4))] + AppellF1[5, 1 + p, -p, 6, E^(2*a)*x^(1/4), -(E^(2*a)*x^(1/4))]))

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Maple [F]
time = 0.85, size = 0, normalized size = 0.00 \[\int \coth ^{p}\left (a +\frac {\ln \left (x \right )}{8}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+1/8*ln(x))^p,x)

[Out]

int(coth(a+1/8*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+1/8*log(x))^p,x, algorithm="maxima")

[Out]

integrate(coth(a + 1/8*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(coth(a+1/8*log(x))^p,x, algorithm="fricas")

[Out]

integral(coth(a + 1/8*log(x))^p, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \coth ^{p}{\left (a + \frac {\log {\left (x \right )}}{8} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+1/8*ln(x))**p,x)

[Out]

Integral(coth(a + log(x)/8)**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+1/8*log(x))^p,x, algorithm="giac")

[Out]

integrate(coth(a + 1/8*log(x))^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (a+\frac {\ln \left (x\right )}{8}\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + log(x)/8)^p,x)

[Out]

int(coth(a + log(x)/8)^p, x)

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