3.97 \(\int \frac {\text {sech}^3(x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=6 \[ \text {sech}(x)+\tan ^{-1}(\sinh (x)) \]

[Out]

arctan(sinh(x))+sech(x)

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Rubi [A]  time = 0.04, antiderivative size = 6, 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 = {3501, 3770} \[ \text {sech}(x)+\tan ^{-1}(\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^3/(1 + Tanh[x]),x]

[Out]

ArcTan[Sinh[x]] + Sech[x]

Rule 3501

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d^2*
(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n -
1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\text {sech}^3(x)}{1+\tanh (x)} \, dx &=\text {sech}(x)+\int \text {sech}(x) \, dx\\ &=\tan ^{-1}(\sinh (x))+\text {sech}(x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 12, normalized size = 2.00 \[ \text {sech}(x)+2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^3/(1 + Tanh[x]),x]

[Out]

2*ArcTan[Tanh[x/2]] + Sech[x]

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fricas [B]  time = 0.42, size = 48, normalized size = 8.00 \[ \frac {2 \, {\left ({\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + \cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(1+tanh(x)),x, algorithm="fricas")

[Out]

2*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*arctan(cosh(x) + sinh(x)) + cosh(x) + sinh(x))/(cosh(x)^2 +
2*cosh(x)*sinh(x) + sinh(x)^2 + 1)

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giac [B]  time = 0.14, size = 18, normalized size = 3.00 \[ \frac {2 \, e^{x}}{e^{\left (2 \, x\right )} + 1} + 2 \, \arctan \left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(1+tanh(x)),x, algorithm="giac")

[Out]

2*e^x/(e^(2*x) + 1) + 2*arctan(e^x)

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maple [B]  time = 0.08, size = 21, normalized size = 3.50 \[ \frac {2}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^3/(1+tanh(x)),x)

[Out]

2/(tanh(1/2*x)^2+1)+2*arctan(tanh(1/2*x))

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maxima [B]  time = 0.41, size = 22, normalized size = 3.67 \[ \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} + 1} - 2 \, \arctan \left (e^{\left (-x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(1+tanh(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.04, size = 18, normalized size = 3.00 \[ 2\,\mathrm {atan}\left ({\mathrm {e}}^x\right )+\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^3*(tanh(x) + 1)),x)

[Out]

2*atan(exp(x)) + (2*exp(x))/(exp(2*x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{3}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**3/(1+tanh(x)),x)

[Out]

Integral(sech(x)**3/(tanh(x) + 1), x)

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