### 3.648 $$\int (a \coth (x)+b \text{csch}(x)) \, dx$$

Optimal. Leaf size=12 $a \log (\sinh (x))-b \tanh ^{-1}(\cosh (x))$

[Out]

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

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Rubi [A]  time = 0.0104265, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {3475, 3770} $a \log (\sinh (x))-b \tanh ^{-1}(\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[a*Coth[x] + b*Csch[x],x]

[Out]

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

Rule 3475

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

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

Mathematica [A]  time = 0.0037515, size = 15, normalized size = 1.25 $a \log (\sinh (x))+b \log \left (\tanh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[a*Coth[x] + b*Csch[x],x]

[Out]

a*Log[Sinh[x]] + b*Log[Tanh[x/2]]

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Maple [A]  time = 0.002, size = 14, normalized size = 1.2 \begin{align*} a\ln \left ( \sinh \left ( x \right ) \right ) +b\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(a*coth(x)+b*csch(x),x)

[Out]

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

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Maxima [A]  time = 1.05417, size = 18, normalized size = 1.5 \begin{align*} a \log \left (\sinh \left (x\right )\right ) + b \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(sinh(x)) + b*log(tanh(1/2*x))

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Fricas [B]  time = 2.10259, size = 108, normalized size = 9. \begin{align*} -a x +{\left (a - b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (a + b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a*coth(x) + b*csch(x), x)

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Giac [B]  time = 1.14124, size = 45, normalized size = 3.75 \begin{align*} -a{\left (x - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )\right )} - b{\left (\log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(x - log(abs(e^(2*x) - 1))) - b*(log(e^x + 1) - log(abs(e^x - 1)))