3.1.94 \(\int \frac {\text {csch}^2(x)}{1+\coth (x)} \, dx\) [94]

Optimal. Leaf size=7 \[ -\log (1+\coth (x)) \]

[Out]

-ln(1+coth(x))

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Rubi [A]
time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3568, 31} \begin {gather*} -\log (\coth (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(1 + Coth[x]),x]

[Out]

-Log[1 + Coth[x]]

Rule 31

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

Rule 3568

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} -x+\log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^2/(1 + Coth[x]),x]

[Out]

-x + Log[Sinh[x]]

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Maple [A]
time = 0.50, size = 8, normalized size = 1.14

method	result	size
derivativedivides	\(-\ln \left (1+\coth \left (x \right )\right )\)	\(8\)
default	\(-\ln \left (1+\coth \left (x \right )\right )\)	\(8\)
risch	\(-2 x +\ln \left ({\mathrm e}^{2 x}-1\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^2/(1+coth(x)),x,method=_RETURNVERBOSE)

[Out]

-ln(1+coth(x))

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Maxima [A]
time = 0.26, size = 7, normalized size = 1.00 \begin {gather*} -\log \left (\coth \left (x\right ) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(1+coth(x)),x, algorithm="maxima")

[Out]

-log(coth(x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (7) = 14\).
time = 0.42, size = 18, normalized size = 2.57 \begin {gather*} -2 \, x + \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(1+coth(x)),x, algorithm="fricas")

[Out]

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

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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 )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**2/(1+coth(x)),x)

[Out]

Integral(csch(x)**2/(coth(x) + 1), x)

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Giac [A]
time = 0.41, size = 12, normalized size = 1.71 \begin {gather*} -2 \, x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^2/(1+coth(x)),x, algorithm="giac")

[Out]

-2*x + log(abs(e^(2*x) - 1))

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Mupad [B]
time = 1.18, size = 11, normalized size = 1.57 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(coth(x) + 1)),x)

[Out]

log(exp(2*x) - 1) - 2*x

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