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

Optimal. Leaf size=17 \[ a x-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d} \]

[Out]

a*x - (b*ArcTanh[Cosh[c + d*x]])/d

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Rubi [A]  time = 0.0101213, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3770} \[ a x-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Csch[c + d*x],x]

[Out]

a*x - (b*ArcTanh[Cosh[c + d*x]])/d

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

Mathematica [B]  time = 0.0139126, size = 43, normalized size = 2.53 \[ a x+\frac{b \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.003, size = 20, normalized size = 1.2 \begin{align*} ax+{\frac{b}{d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.982444, size = 26, normalized size = 1.53 \begin{align*} a x + \frac{b \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.53912, size = 131, normalized size = 7.71 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*csch(c + d*x), x)

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Giac [B]  time = 1.15953, size = 49, normalized size = 2.88 \begin{align*} a x - b{\left (\frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{d} - \frac{\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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