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

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 = 0.0058973, 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 = 0.0127889, 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 [A]  time = 0.007, size = 13, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({\rm csch} \left (bx+a\right ) \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)*csch(b*x+a),x)

[Out]

-1/b*ln(csch(b*x+a))

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Maxima [B]  time = 1.09975, size = 31, normalized size = 2.82 \begin{align*} \frac{\log \left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e^(b*x + a) - e^(-b*x - a))/b

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Fricas [B]  time = 2.07281, 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(cosh(b*x+a)*csch(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x \right )} \operatorname{csch}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*csch(b*x+a),x)

[Out]

Integral(cosh(a + b*x)*csch(a + b*x), x)

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Giac [B]  time = 1.21458, 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(cosh(b*x+a)*csch(b*x+a),x, algorithm="giac")

[Out]

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