3.7 \(\int \coth (a+b x) \, dx\)

Optimal. Leaf size=11 \[ \frac{\log (\sinh (a+b x))}{b} \]

[Out]

Log[Sinh[a + b*x]]/b

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Rubi [A]  time = 0.0059246, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3475} \[ \frac{\log (\sinh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Coth[a + b*x],x]

[Out]

Log[Sinh[a + b*x]]/b

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \coth (a+b x) \, dx &=\frac{\log (\sinh (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0094081, size = 19, normalized size = 1.73 \[ \frac{\log (\tanh (a+b x))+\log (\cosh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + b*x],x]

[Out]

(Log[Cosh[a + b*x]] + Log[Tanh[a + b*x]])/b

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Maple [B]  time = 0.002, size = 30, normalized size = 2.7 \begin{align*} -{\frac{\ln \left ({\rm coth} \left (bx+a\right )-1 \right ) }{2\,b}}-{\frac{\ln \left ({\rm coth} \left (bx+a\right )+1 \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(b*x+a),x)

[Out]

-1/2/b*ln(coth(b*x+a)-1)-1/2/b*ln(coth(b*x+a)+1)

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Maxima [A]  time = 1.02564, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (\sinh \left (b x + a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a),x, algorithm="maxima")

[Out]

log(sinh(b*x + a))/b

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Fricas [B]  time = 2.44702, size = 88, normalized size = 8. \begin{align*} -\frac{b x - \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.440701, size = 27, normalized size = 2.45 \begin{align*} \begin{cases} x - \frac{\log{\left (\tanh{\left (a + b x \right )} + 1 \right )}}{b} + \frac{\log{\left (\tanh{\left (a + b x \right )} \right )}}{b} & \text{for}\: b \neq 0 \\x \coth{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a),x)

[Out]

Piecewise((x - log(tanh(a + b*x) + 1)/b + log(tanh(a + b*x))/b, Ne(b, 0)), (x*coth(a), True))

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Giac [B]  time = 1.18339, size = 38, normalized size = 3.45 \begin{align*} -\frac{b x + a}{b} + \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a),x, algorithm="giac")

[Out]

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