3.2.80 \(\int \coth (d (a+b \log (c x^n))) \, dx\) [180]

Optimal. Leaf size=52 \[ x-2 x \, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

x-2*x*hypergeom([1, 1/2/b/d/n],[1+1/2/b/d/n],exp(2*a*d)*(c*x^n)^(2*b*d))

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Rubi [A]
time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5655, 5657, 470, 371} \begin {gather*} x-2 x \, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[d*(a + b*Log[c*x^n])],x]

[Out]

x - 2*x*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

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

Rule 5655

Int[Coth[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Coth[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5657

Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*((-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, e, m, p}, x]

Rubi steps

\begin {align*} \int \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(198\) vs. \(2(52)=104\).
time = 6.16, size = 198, normalized size = 3.81 \begin {gather*} -\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} x \, _2F_1\left (1,1+\frac {1}{2 b d n};2+\frac {1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{1+2 b d n}-x \left (\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+\, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text {csch}\left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sinh (b d n \log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])],x]

[Out]

-((E^(2*d*(a + b*Log[c*x^n]))*x*Hypergeometric2F1[1, 1 + 1/(2*b*d*n), 2 + 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n
]))])/(1 + 2*b*d*n)) - x*(Coth[d*(a + b*Log[c*x^n])] - Coth[d*(a - b*n*Log[x] + b*Log[c*x^n])] + Hypergeometri
c2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))] + Csch[d*(a + b*Log[c*x^n])]*Csch[d*(a - b*n
*Log[x] + b*Log[c*x^n])]*Sinh[b*d*n*Log[x]])

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Maple [F]
time = 0.66, size = 0, normalized size = 0.00 \[\int \coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a+b*ln(c*x^n))),x)

[Out]

int(coth(d*(a+b*ln(c*x^n))),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(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

x - integrate(1/(c^(b*d)*e^(b*d*log(x^n) + a*d) + 1), x) + integrate(1/(c^(b*d)*e^(b*d*log(x^n) + a*d) - 1), 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(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral(coth(b*d*log(c*x^n) + a*d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*ln(c*x**n))),x)

[Out]

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

[Out]

integrate(coth((b*log(c*x^n) + a)*d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a + b*log(c*x^n))),x)

[Out]

int(coth(d*(a + b*log(c*x^n))), x)

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