3.247 \(\int \coth (x) \, dx\)

Optimal. Leaf size=3 \[ \log (\sinh (x)) \]

[Out]

Log[Sinh[x]]

_______________________________________________________________________________________

Rubi [A]  time = 0.00608192, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]  Int[Coth[x],x]

[Out]

Log[Sinh[x]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.1172, size = 17, normalized size = 5.67 \[ - \frac{\log{\left (- \tanh ^{2}{\left (x \right )} + 1 \right )}}{2} + \frac{\log{\left (\tanh ^{2}{\left (x \right )} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(coth(x),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00534756, size = 3, normalized size = 1. \[ \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Coth[x],x]

[Out]

Log[Sinh[x]]

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 4, normalized size = 1.3 \[ \ln \left ( \sinh \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(coth(x),x)

[Out]

ln(sinh(x))

_______________________________________________________________________________________

Maxima [A]  time = 1.46243, size = 4, normalized size = 1.33 \[ \log \left (\sinh \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(coth(x),x, algorithm="maxima")

[Out]

log(sinh(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.221777, size = 24, normalized size = 8. \[ -x + \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(coth(x),x, algorithm="fricas")

[Out]

-x + log(2*sinh(x)/(cosh(x) - sinh(x)))

_______________________________________________________________________________________

Sympy [A]  time = 0.401351, size = 12, normalized size = 4. \[ x - \log{\left (\tanh{\left (x \right )} + 1 \right )} + \log{\left (\tanh{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(coth(x),x)

[Out]

x - log(tanh(x) + 1) + log(tanh(x))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227696, size = 16, normalized size = 5.33 \[ -x +{\rm ln}\left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(coth(x),x, algorithm="giac")

[Out]

-x + ln(abs(e^(2*x) - 1))