\(\int (a+b \text {csch}(c+d x)) \, dx\) [73]

Optimal result

Integrand size = 10, antiderivative size = 17 \[ \int (a+b \text {csch}(c+d x)) \, dx=a x-\frac {b \text {arctanh}(\cosh (c+d x))}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3855} \[ \int (a+b \text {csch}(c+d x)) \, dx=a x-\frac {b \text {arctanh}(\cosh (c+d x))}{d} \]

[In]

Int[a + b*Csch[c + d*x],x]

[Out]

a*x - (b*ArcTanh[Cosh[c + d*x]])/d

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}& = a x+b \int \text {csch}(c+d x) \, dx \\ & = a x-\frac {b \text {arctanh}(\cosh (c+d x))}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(17)=34\).

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.53 \[ \int (a+b \text {csch}(c+d x)) \, dx=a x-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \]

[In]

Integrate[a + b*Csch[c + d*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18

method	result	size
default	\(a x +\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(20\)
parallelrisch	\(a x +\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(20\)
parts	\(a x +\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(20\)
derivativedivides	\(\frac {a \left (d x +c \right )+b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(25\)
risch	\(a x +\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(34\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int (a+b \text {csch}(c+d x)) \, dx=\frac {a d x - b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{d} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int (a+b \text {csch}(c+d x)) \, dx=a x + b \left (\begin {cases} \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{d} & \text {for}\: d \neq 0 \\x \operatorname {csch}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int (a+b \text {csch}(c+d x)) \, dx=a x + \frac {b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]

[In]

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

[Out]

a*x + b*log(tanh(1/2*d*x + 1/2*c))/d

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88 \[ \int (a+b \text {csch}(c+d x)) \, dx=a x - \frac {b {\left (\log \left (e^{\left (d x + c\right )} + 1\right ) - \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )\right )}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.47 \[ \int (a+b \text {csch}(c+d x)) \, dx=a\,x-\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]

[In]

int(a + b/sinh(c + d*x),x)

[Out]

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