3.2.10 \(\int \frac {\text {sech}^2(x)}{1+\coth (x)} \, dx\) [110]

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

[Out]

-ln(1+coth(x))-ln(tanh(x))+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*} \tanh (x)-\log (\tanh (x))-\log (\coth (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/(1 + Coth[x]),x]

[Out]

-Log[1 + Coth[x]] - Log[Tanh[x]] + 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 {sech}^2(x)}{1+\coth (x)} \, dx &=-\text {Subst}\left (\int \frac {1}{x^2 (1+x)} \, dx,x,\coth (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,\coth (x)\right )\\ &=-\log (1+\coth (x))-\log (\tanh (x))+\tanh (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 9, normalized size = 0.60 \begin {gather*} -x+\log (\cosh (x))+\tanh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^2/(1 + Coth[x]),x]

[Out]

-x + Log[Cosh[x]] + Tanh[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(15)=30\).
time = 0.58, size = 36, normalized size = 2.40

method	result	size
risch	\(-2 x -\frac {2}{1+{\mathrm e}^{2 x}}+\ln \left (1+{\mathrm e}^{2 x}\right )\)	\(22\)
default	\(-2 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {2 \tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 18, normalized size = 1.20 \begin {gather*} \frac {2}{e^{\left (-2 \, x\right )} + 1} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(e^(-2*x) + 1) + log(e^(-2*x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (15) = 30\).
time = 0.35, size = 78, normalized size = 5.20 \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 \, \cosh \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(sech(x)^2/(1+coth(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*
cosh(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 {sech}^{2}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(x)**2/(coth(x) + 1), x)

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Giac [A]
time = 0.41, size = 27, normalized size = 1.80 \begin {gather*} -2 \, x - \frac {e^{\left (2 \, x\right )} + 3}{e^{\left (2 \, x\right )} + 1} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.19, size = 21, normalized size = 1.40 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}+1\right )-2\,x-\frac {2}{{\mathrm {e}}^{2\,x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2*(coth(x) + 1)),x)

[Out]

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

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