3.1.74 \(\int \frac {\text {csch}^2(x)}{1+\tanh (x)} \, dx\) [74]

Optimal. Leaf size=15 \[ -\coth (x)-\log (\tanh (x))+\log (1+\tanh (x)) \]

[Out]

-coth(x)-ln(tanh(x))+ln(1+tanh(x))

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Rubi [A]
time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3597, 46} \begin {gather*} -\coth (x)-\log (\tanh (x))+\log (\tanh (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^2/(1 + Tanh[x]),x]

[Out]

-Coth[x] - Log[Tanh[x]] + Log[1 + Tanh[x]]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 3597

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

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 11, normalized size = 0.73 \begin {gather*} x-\coth (x)-\log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^2/(1 + Tanh[x]),x]

[Out]

x - Coth[x] - Log[Sinh[x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(15)=30\).
time = 0.50, size = 32, normalized size = 2.13

method	result	size
risch	\(2 x -\frac {2}{{\mathrm e}^{2 x}-1}-\ln \left ({\mathrm e}^{2 x}-1\right )\)	\(24\)
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\ln \left (\tanh \left (\frac {x}{2}\right )\right )+2 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*tanh(1/2*x)-1/2/tanh(1/2*x)-ln(tanh(1/2*x))+2*ln(tanh(1/2*x)+1)

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Maxima [A]
time = 0.26, size = 29, normalized size = 1.93 \begin {gather*} \frac {2}{e^{\left (-2 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (15) = 30\).
time = 0.37, size = 77, normalized size = 5.13 \begin {gather*} \frac {2 \, x \cosh \left (x\right )^{2} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, x \sinh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \, x - 2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*cosh(x)^2 + 4*x*cosh(x)*sinh(x) + 2*x*sinh(x)^2 - (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*log(2*s
inh(x)/(cosh(x) - sinh(x))) - 2*x - 2)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(x)**2/(tanh(x) + 1), x)

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Giac [A]
time = 0.41, size = 29, normalized size = 1.93 \begin {gather*} 2 \, x + \frac {e^{\left (2 \, x\right )} - 3}{e^{\left (2 \, x\right )} - 1} - \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+tanh(x)),x, algorithm="giac")

[Out]

2*x + (e^(2*x) - 3)/(e^(2*x) - 1) - log(abs(e^(2*x) - 1))

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Mupad [B]
time = 1.08, size = 23, normalized size = 1.53 \begin {gather*} 2\,x-\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {2}{{\mathrm {e}}^{2\,x}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2*(tanh(x) + 1)),x)

[Out]

2*x - log(exp(2*x) - 1) - 2/(exp(2*x) - 1)

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