3.1.97 \(\int \frac {\text {sech}^3(x)}{1+\tanh (x)} \, dx\) [97]

Optimal. Leaf size=6 \[ \text {ArcTan}(\sinh (x))+\text {sech}(x) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 = {3582, 3855} \begin {gather*} \text {ArcTan}(\sinh (x))+\text {sech}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x]^3/(1 + Tanh[x]),x]

[Out]

ArcTan[Sinh[x]] + Sech[x]

Rule 3582

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 3855

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 = 0.02, size = 12, normalized size = 2.00 \begin {gather*} 2 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+\text {sech}(x) \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(20\) vs. \(2(6)=12\).
time = 0.50, size = 21, normalized size = 3.50

method	result	size
default	\(\frac {2}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(21\)
risch	\(\frac {2 \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}+i \ln \left ({\mathrm e}^{x}+i\right )-i \ln \left ({\mathrm e}^{x}-i\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (6) = 12\).
time = 0.48, size = 22, normalized size = 3.67 \begin {gather*} \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} + 1} - 2 \, \arctan \left (e^{\left (-x\right )}\right ) \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (6) = 12\).
time = 0.35, size = 48, normalized size = 8.00 \begin {gather*} \frac {2 \, {\left ({\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\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)^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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (6) = 12\).
time = 0.41, size = 18, normalized size = 3.00 \begin {gather*} \frac {2 \, e^{x}}{e^{\left (2 \, x\right )} + 1} + 2 \, \arctan \left (e^{x}\right ) \end {gather*}

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

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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