\(\int \text {csch}(a+b x) \, dx\) [1]

Optimal result

Integrand size = 6, antiderivative size = 12 \[ \int \text {csch}(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3855} \[ \int \text {csch}(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{b} \]

[In]

Int[Csch[a + b*x],x]

[Out]

-(ArcTanh[Cosh[a + b*x]]/b)

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arctanh}(\cosh (a+b x))}{b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 3.17 \[ \int \text {csch}(a+b x) \, dx=-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}+\frac {\log \left (\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \]

[In]

Integrate[Csch[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\)	\(15\)
default	\(\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\)	\(15\)
parallelrisch	\(\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\)	\(15\)
risch	\(\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b}\)	\(29\)

[In]

int(csch(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 3.17 \[ \int \text {csch}(a+b x) \, dx=-\frac {\log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b} \]

[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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.42 \[ \int \text {csch}(a+b x) \, dx=\begin {cases} \frac {\log {\left (\tanh {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b} & \text {for}\: b \neq 0 \\x \operatorname {csch}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((log(tanh(a/2 + b*x/2))/b, Ne(b, 0)), (x*csch(a), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \text {csch}(a+b x) \, dx=\frac {\log \left (\tanh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}{b} \]

[In]

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

[Out]

log(tanh(1/2*b*x + 1/2*a))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \text {csch}(a+b x) \, dx=-\frac {\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \text {csch}(a+b x) \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]

[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)