∫ tanh(x)_____∘ 1 + tanh(x) dx [127]

Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\tanh (x)}} \]

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3607, 3561, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {\tanh (x)+1}} \end {gather*}

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq

rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(

b*c - a*d))*((a + b*Tan[e + f*x])^m/(2*a*f*m)), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1

), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 30, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\tanh (x)}} \end {gather*}

________________________________________________________________________________________


method	result	size

derivativedivides	\(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\tanh \left (x \right )}}\)	\(25\)
default	\(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\tanh \left (x \right )}}\)	\(25\)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (24) = 48\).
time = 0.34, size = 85, normalized size = 2.83 \begin {gather*} \frac {{\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right ) + 4 \, \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}}{4 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}

1/4*((sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*log(-2*sqrt(2)*sqrt(cosh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x))

- 2*cosh(x)^2 - 4*cosh(x)*sinh(x) - 2*sinh(x)^2 - 1) + 4*sqrt(cosh(x)/(cosh(x) - sinh(x))))/(cosh(x) + sinh(x)

________________________________________________________________________________________

Sympy [A]
time = 1.56, size = 63, normalized size = 2.10 \begin {gather*} - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {\tanh {\left (x \right )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\left (x \right )} > 1 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {\tanh {\left (x \right )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\left (x \right )} < 1 \end {cases} + \frac {1}{\sqrt {\tanh {\left (x \right )} + 1}} \end {gather*}

-Piecewise((-sqrt(2)*acoth(sqrt(2)*sqrt(tanh(x) + 1)/2)/2, tanh(x) > 1), (-sqrt(2)*atanh(sqrt(2)*sqrt(tanh(x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
time = 0.41, size = 52, normalized size = 1.73 \begin {gather*} \frac {1}{4} \, \sqrt {2} {\left (\frac {2}{\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}} - \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \end {gather*}

1/4*sqrt(2)*(2/(sqrt(e^(4*x) + e^(2*x)) - e^(2*x)) - log(-2*sqrt(e^(4*x) + e^(2*x)) + 2*e^(2*x) + 1))

________________________________________________________________________________________

Mupad [B]
time = 0.13, size = 24, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )}{2}+\frac {1}{\sqrt {\mathrm {tanh}\left (x\right )+1}} \end {gather*}

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

________________________________________________________________________________________

3.2.27 \(\int \frac {\tanh (x)}{\sqrt {1+\tanh (x)}} \, dx\) [127]