### 3.1019 $$\int \frac{\text{csch}^2(x)}{a+b \coth (x)} \, dx$$

Optimal. Leaf size=12 $-\frac{\log (a+b \coth (x))}{b}$

[Out]

-(Log[a + b*Coth[x]]/b)

________________________________________________________________________________________

Rubi [A]  time = 0.0484419, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {3506, 31} $-\frac{\log (a+b \coth (x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[x]^2/(a + b*Coth[x]),x]

[Out]

-(Log[a + b*Coth[x]]/b)

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

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{\text{csch}^2(x)}{a+b \coth (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \coth (x)\right )}{b}\\ &=-\frac{\log (a+b \coth (x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.0536766, size = 20, normalized size = 1.67 $\frac{\log (\sinh (x))-\log (a \sinh (x)+b \cosh (x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[x]^2/(a + b*Coth[x]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 13, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( a+b{\rm coth} \left (x\right ) \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^2/(a+b*coth(x)),x)

[Out]

-ln(a+b*coth(x))/b

________________________________________________________________________________________

Maxima [A]  time = 1.00074, size = 16, normalized size = 1.33 \begin{align*} -\frac{\log \left (b \coth \left (x\right ) + a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(a+b*coth(x)),x, algorithm="maxima")

[Out]

-log(b*coth(x) + a)/b

________________________________________________________________________________________

Fricas [B]  time = 2.16145, size = 127, normalized size = 10.58 \begin{align*} -\frac{\log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(a+b*coth(x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{2}{\left (x \right )}}{a + b \coth{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**2/(a+b*coth(x)),x)

[Out]

Integral(csch(x)**2/(a + b*coth(x)), x)

________________________________________________________________________________________

Giac [B]  time = 1.09929, size = 62, normalized size = 5.17 \begin{align*} -\frac{{\left (a + b\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a b + b^{2}} + \frac{\log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(a+b*coth(x)),x, algorithm="giac")

[Out]

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