3.5.25 \(\int \frac {\coth (x)}{1-\sinh ^2(x)} \, dx\) [425]

Optimal. Leaf size=17 \[ \log (\sinh (x))-\frac {1}{2} \log \left (1-\sinh ^2(x)\right ) \]

[Out]

ln(sinh(x))-1/2*ln(1-sinh(x)^2)

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Rubi [A]
time = 0.02, antiderivative size = 17, 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 = {3273, 36, 31, 29} \begin {gather*} \log (\sinh (x))-\frac {1}{2} \log \left (1-\sinh ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(1 - Sinh[x]^2),x]

[Out]

Log[Sinh[x]] - Log[1 - Sinh[x]^2]/2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 3273

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m
+ 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\coth (x)}{1-\sinh ^2(x)} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,\sinh ^2(x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sinh ^2(x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sinh ^2(x)\right )\\ &=\log (\sinh (x))-\frac {1}{2} \log \left (1-\sinh ^2(x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.35 \begin {gather*} -2 \left (-\frac {1}{2} \log (\sinh (x))+\frac {1}{4} \log \left (1-\sinh ^2(x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(1 - Sinh[x]^2),x]

[Out]

-2*(-1/2*Log[Sinh[x]] + Log[1 - Sinh[x]^2]/4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(15)=30\).
time = 0.61, size = 41, normalized size = 2.41

method	result	size
risch	\(\ln \left ({\mathrm e}^{2 x}-1\right )-\frac {\ln \left ({\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1\right )}{2}\)	\(24\)
default	\(-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )-2 \tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+2 \tanh \left (\frac {x}{2}\right )-1\right )}{2}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(1-sinh(x)^2),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(tanh(1/2*x)^2-2*tanh(1/2*x)-1)-1/2*ln(tanh(1/2*x)^2+2*tanh(1/2*x)-1)+ln(tanh(1/2*x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (15) = 30\).
time = 0.26, size = 45, normalized size = 2.65 \begin {gather*} -\frac {1}{2} \, \log \left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) - \frac {1}{2} \, \log \left (-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(1-sinh(x)^2),x, algorithm="maxima")

[Out]

-1/2*log(2*e^(-x) + e^(-2*x) - 1) + log(e^(-x) + 1) + log(e^(-x) - 1) - 1/2*log(-2*e^(-x) + e^(-2*x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (15) = 30\).
time = 0.48, size = 47, normalized size = 2.76 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {2 \, {\left (\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(1-sinh(x)^2),x, algorithm="fricas")

[Out]

-1/2*log(2*(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + log(2*sinh(x)/(cosh(x) -
sinh(x)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\coth {\left (x \right )}}{\sinh ^{2}{\left (x \right )} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(1-sinh(x)**2),x)

[Out]

-Integral(coth(x)/(sinh(x)**2 - 1), x)

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Giac [A]
time = 0.43, size = 25, normalized size = 1.47 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(1-sinh(x)^2),x, algorithm="giac")

[Out]

-1/2*log(abs(e^(4*x) - 6*e^(2*x) + 1)) + log(abs(e^(2*x) - 1))

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Mupad [B]
time = 0.08, size = 27, normalized size = 1.59 \begin {gather*} \ln \left (5184\,{\mathrm {e}}^{2\,x}-5184\right )-\frac {\ln \left (9\,{\mathrm {e}}^{4\,x}-54\,{\mathrm {e}}^{2\,x}+9\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-coth(x)/(sinh(x)^2 - 1),x)

[Out]

log(5184*exp(2*x) - 5184) - log(9*exp(4*x) - 54*exp(2*x) + 9)/2

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