### 3.649 $$\int \frac{1}{a \coth (x)+b \text{csch}(x)} \, dx$$

Optimal. Leaf size=11 $\frac{\log (a \cosh (x)+b)}{a}$

[Out]

Log[b + a*Cosh[x]]/a

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Rubi [A]  time = 0.0462933, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {3160, 2668, 31} $\frac{\log (a \cosh (x)+b)}{a}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*Coth[x] + b*Csch[x])^(-1),x]

[Out]

Log[b + a*Cosh[x]]/a

Rule 3160

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(-1), x_Symbol] :> Int[Sin[d + e*x
]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{a \coth (x)+b \text{csch}(x)} \, dx &=i \int \frac{\sinh (x)}{i b+i a \cosh (x)} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{i b+x} \, dx,x,i a \cosh (x)\right )}{a}\\ &=\frac{\log (b+a \cosh (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0179501, size = 11, normalized size = 1. $\frac{\log (a \cosh (x)+b)}{a}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*Coth[x] + b*Csch[x])^(-1),x]

[Out]

Log[b + a*Cosh[x]]/a

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Maple [B]  time = 0.034, size = 51, normalized size = 4.6 \begin{align*} -{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{a}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*coth(x)+b*csch(x)),x)

[Out]

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

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Maxima [B]  time = 1.03671, size = 35, normalized size = 3.18 \begin{align*} \frac{x}{a} + \frac{\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.06924, size = 72, normalized size = 6.55 \begin{align*} -\frac{x - \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x)),x, algorithm="fricas")

[Out]

-(x - log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a \coth{\left (x \right )} + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x)),x)

[Out]

Integral(1/(a*coth(x) + b*csch(x)), x)

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Giac [A]  time = 1.12907, size = 26, normalized size = 2.36 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*coth(x)+b*csch(x)),x, algorithm="giac")

[Out]

log(abs(a*(e^(-x) + e^x) + 2*b))/a