3.1 \(\int \text{csch}(a+b x) \, dx\)

Optimal. Leaf size=12 \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]

[Out]

-(ArcTanh[Cosh[a + b*x]]/b)

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Rubi [A]  time = 0.0062549, antiderivative size = 12, 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 = {3770} \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*x],x]

[Out]

-(ArcTanh[Cosh[a + b*x]]/b)

Rule 3770

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

Rubi steps

\begin{align*} \int \text{csch}(a+b x) \, dx &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{b}\\ \end{align*}

Mathematica [B]  time = 0.0176559, size = 38, normalized size = 3.17 \[ \frac{\log \left (\sinh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*x],x]

[Out]

-(Log[Cosh[a/2 + (b*x)/2]]/b) + Log[Sinh[a/2 + (b*x)/2]]/b

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Maple [A]  time = 0.003, size = 15, normalized size = 1.3 \begin{align*}{\frac{1}{b}\ln \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*ln(tanh(1/2*b*x+1/2*a))

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Maxima [A]  time = 1.00528, size = 19, normalized size = 1.58 \begin{align*} \frac{\log \left (\tanh \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tanh(1/2*b*x + 1/2*a))/b

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Fricas [B]  time = 1.55341, size = 116, normalized size = 9.67 \begin{align*} -\frac{\log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(cosh(b*x + a) + sinh(b*x + a) + 1) - log(cosh(b*x + a) + sinh(b*x + a) - 1))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(a + b*x), x)

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Giac [B]  time = 1.14747, size = 39, normalized size = 3.25 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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