3.3.20 \(\int \log (a \text {csch}^n(x)) \, dx\) [220]

Optimal. Leaf size=43 \[ -\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right ) \]

[Out]

-1/2*n*x^2+n*x*ln(1-exp(2*x))+x*ln(a*csch(x)^n)+1/2*n*polylog(2,exp(2*x))

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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3797, 2221, 2317, 2438} \begin {gather*} \frac {1}{2} n \text {PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[a*Csch[x]^n],x]

[Out]

-1/2*(n*x^2) + n*x*Log[1 - E^(2*x)] + x*Log[a*Csch[x]^n] + (n*PolyLog[2, E^(2*x)])/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2221

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \log \left (a \text {csch}^n(x)\right ) \, dx &=x \log \left (a \text {csch}^n(x)\right )+\int n x \coth (x) \, dx\\ &=x \log \left (a \text {csch}^n(x)\right )+n \int x \coth (x) \, dx\\ &=-\frac {n x^2}{2}+x \log \left (a \text {csch}^n(x)\right )-(2 n) \int \frac {e^{2 x} x}{1-e^{2 x}} \, dx\\ &=-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-n \int \log \left (1-e^{2 x}\right ) \, dx\\ &=-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-\frac {1}{2} n \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )\\ &=-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} \frac {n x^2}{2}+n x \log \left (1-e^{-2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (e^{-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[a*Csch[x]^n],x]

[Out]

(n*x^2)/2 + n*x*Log[1 - E^(-2*x)] + x*Log[a*Csch[x]^n] - (n*PolyLog[2, E^(-2*x)])/2

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \ln \left (a \mathrm {csch}\left (x \right )^{n}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(a*csch(x)^n),x)

[Out]

int(ln(a*csch(x)^n),x)

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Maxima [A]
time = 0.34, size = 47, normalized size = 1.09 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) - 2 \, {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_2\left (e^{x}\right )\right )} n + x \log \left (a \operatorname {csch}\left (x\right )^{n}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*csch(x)^n),x, algorithm="maxima")

[Out]

-1/2*(x^2 - 2*x*log(e^x + 1) - 2*x*log(-e^x + 1) - 2*dilog(-e^x) - 2*dilog(e^x))*n + x*log(a*csch(x)^n)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (36) = 72\).
time = 0.41, size = 84, normalized size = 1.95 \begin {gather*} -\frac {1}{2} \, n x^{2} + n x \log \left (\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}\right ) + n x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + n x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + n {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + n {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) + x \log \left (a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(a*csch(x)^n),x, algorithm="fricas")

[Out]

-1/2*n*x^2 + n*x*log(2*(cosh(x) + sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)) + n*x*log(cosh(x)
+ sinh(x) + 1) + n*x*log(-cosh(x) - sinh(x) + 1) + n*dilog(cosh(x) + sinh(x)) + n*dilog(-cosh(x) - sinh(x)) +
x*log(a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(a*csch(x)**n),x)

[Out]

Integral(log(a*csch(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(log(a*csch(x)^n),x, algorithm="giac")

[Out]

integrate(log(a*csch(x)^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a*(1/sinh(x))^n),x)

[Out]

int(log(a*(1/sinh(x))^n), x)

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