3.669 \(\int \frac {1}{-\coth (x)+\text {csch}(x)} \, dx\)

Optimal. Leaf size=9 \[ -\log (1-\cosh (x)) \]

[Out]

-ln(1-cosh(x))

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Rubi [A]  time = 0.03, antiderivative size = 9, 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 = {3160, 2667, 31} \[ -\log (1-\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(-Coth[x] + Csch[x])^(-1),x]

[Out]

-Log[1 - Cosh[x]]

Rule 31

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

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 3160

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(-1), x_Symbol] :> Int[Sin[d + e*x
]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 9, normalized size = 1.00 \[ -2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Coth[x] + Csch[x])^(-1),x]

[Out]

-2*Log[Sinh[x/2]]

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fricas [A]  time = 0.40, size = 11, normalized size = 1.22 \[ x - 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x)),x, algorithm="fricas")

[Out]

x - 2*log(cosh(x) + sinh(x) - 1)

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giac [A]  time = 0.13, size = 10, normalized size = 1.11 \[ x - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x)),x, algorithm="giac")

[Out]

x - 2*log(abs(e^x - 1))

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maple [B]  time = 0.19, size = 23, normalized size = 2.56 \[ \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-coth(x)+csch(x)),x)

[Out]

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

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maxima [A]  time = 0.37, size = 13, normalized size = 1.44 \[ -x - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x)),x, algorithm="maxima")

[Out]

-x - 2*log(e^(-x) - 1)

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mupad [B]  time = 0.04, size = 9, normalized size = 1.00 \[ x-2\,\ln \left ({\mathrm {e}}^x-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(coth(x) - 1/sinh(x)),x)

[Out]

x - 2*log(exp(x) - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{\coth {\relax (x )} - \operatorname {csch}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-coth(x)+csch(x)),x)

[Out]

-Integral(1/(coth(x) - csch(x)), x)

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