3.2.2 \(\int \frac {\text {csch}^2(x)}{a+b \coth (x)} \, dx\) [102]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {3587, 31} \begin {gather*} -\frac {\log (a+b \coth (x))}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(a + b*Coth[x]),x]

[Out]

-(Log[a + b*Coth[x]]/b)

Rule 31

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

Rule 3587

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]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 20, normalized size = 1.67 \begin {gather*} \frac {\log (\sinh (x))-\log (b \cosh (x)+a \sinh (x))}{b} \end {gather*}

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

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Maple [A]
time = 0.57, size = 13, normalized size = 1.08

method	result	size
derivativedivides	\(-\frac {\ln \left (a +b \coth \left (x \right )\right )}{b}\)	\(13\)
default	\(-\frac {\ln \left (a +b \coth \left (x \right )\right )}{b}\)	\(13\)
risch	\(-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{b}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b \coth \left (x\right ) + a\right )}{b} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (12) = 24\).
time = 0.36, size = 43, normalized size = 3.58 \begin {gather*} -\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 {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{2}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \end {gather*}

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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (12) = 24\).
time = 0.42, size = 46, normalized size = 3.83 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.16, size = 51, normalized size = 4.25 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{2\,x}\,\sqrt {-b^2}-a\,\sqrt {-b^2}+b\,{\mathrm {e}}^{2\,x}\,\sqrt {-b^2}}{b^2}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(a + b*coth(x))),x)

[Out]

-(2*atan((a*exp(2*x)*(-b^2)^(1/2) - a*(-b^2)^(1/2) + b*exp(2*x)*(-b^2)^(1/2))/b^2))/(-b^2)^(1/2)

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