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(d*x+c))/d

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [B]  time = 0.01, 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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fricas [B]  time = 0.41, size = 44, normalized size = 2.59 \[ \frac {a d x - b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{d} \]

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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giac [A]  time = 0.11, size = 32, normalized size = 1.88 \[ a x - \frac {b {\left (\log \left (e^{\left (d x + c\right )} + 1\right ) - \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )\right )}}{d} \]

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) - log(abs(e^(d*x + c) - 1)))/d

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maple [A]  time = 0.02, size = 20, normalized size = 1.18 \[ a x +\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d} \]

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 = 0.31, size = 19, normalized size = 1.12 \[ a x + \frac {b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]

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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mupad [B]  time = 0.07, size = 42, normalized size = 2.47 \[ a\,x-\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

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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