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

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

[Out]

-arctanh(cosh(b*x+a))/b

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.02, 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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fricas [B]  time = 0.51, size = 38, normalized size = 3.17 \[ -\frac {\log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b} \]

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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giac [B]  time = 0.13, size = 27, normalized size = 2.25 \[ -\frac {\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]

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) - log(abs(e^(b*x + a) - 1)))/b

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maple [A]  time = 0.02, size = 15, normalized size = 1.25 \[ \frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b} \]

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 = 0.31, size = 14, normalized size = 1.17 \[ \frac {\log \left (\tanh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}{b} \]

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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mupad [B]  time = 0.11, size = 27, normalized size = 2.25 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(a + b*x),x)

[Out]

-(2*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(-b^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}{\left (a + b x \right )}\, dx \]

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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