3.131 \(\int \frac {\tanh ^2(x)}{\sqrt {1+\tanh (x)}} \, dx\)

Optimal. Leaf size=42 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-2 \sqrt {\tanh (x)+1}-\frac {1}{\sqrt {\tanh (x)+1}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3543, 3479, 3480, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-2 \sqrt {\tanh (x)+1}-\frac {1}{\sqrt {\tanh (x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^2/Sqrt[1 + Tanh[x]],x]

[Out]

ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Tanh[x]] - 2*Sqrt[1 + Tanh[x]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 3480

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]

Rule 3543

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
(d^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 37, normalized size = 0.88 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {-2 \tanh (x)-3}{\sqrt {\tanh (x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^2/Sqrt[1 + Tanh[x]],x]

[Out]

ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] + (-3 - 2*Tanh[x])/Sqrt[1 + Tanh[x]]

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fricas [B]  time = 0.82, size = 182, normalized size = 4.33 \[ -\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} + 10 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + 5 \, \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}\right )} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - {\left (\sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + \sqrt {2} \sinh \relax (x)^{3} + {\left (3 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x) + \sqrt {2} \cosh \relax (x)\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} - 2 \, \cosh \relax (x)^{2} - 4 \, \cosh \relax (x) \sinh \relax (x) - 2 \, \sinh \relax (x)^{2} - 1\right )}{4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*(5*sqrt(2)*cosh(x)^2 + 10*sqrt(2)*cosh(x)*sinh(x) + 5*sqrt(2)*sinh(x)^2 + sqrt(2))*sqrt(cosh(x
)/(cosh(x) - sinh(x))) - (sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*sinh(x)^3 + (3*sqrt(2)*cos
h(x)^2 + sqrt(2))*sinh(x) + sqrt(2)*cosh(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))/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*co
sh(x)^2 + 1)*sinh(x) + cosh(x))

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giac [A]  time = 0.18, size = 54, normalized size = 1.29 \[ -\frac {1}{4} \, \sqrt {2} \log \left (-4 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 4 \, e^{\left (2 \, x\right )} + 2\right ) - \frac {5 \, \sqrt {2} e^{\left (2 \, x\right )} + \sqrt {2}}{2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*log(-4*sqrt(e^(4*x) + e^(2*x)) + 4*e^(2*x) + 2) - 1/2*(5*sqrt(2)*e^(2*x) + sqrt(2))/sqrt(e^(4*x)
+ e^(2*x))

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maple [A]  time = 0.08, size = 35, normalized size = 0.83 \[ \frac {\arctanh \left (\frac {\sqrt {1+\tanh \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\tanh \relax (x )}}-2 \sqrt {1+\tanh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/(1+tanh(x))^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)^{2}}{\sqrt {\tanh \relax (x) + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(x)^2/sqrt(tanh(x) + 1), x)

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mupad [B]  time = 0.13, size = 36, normalized size = 0.86 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\relax (x)+1}}{2}\right )}{2}-\frac {3}{\sqrt {\mathrm {tanh}\relax (x)+1}}-\frac {2\,\mathrm {tanh}\relax (x)}{\sqrt {\mathrm {tanh}\relax (x)+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^2/(tanh(x) + 1)^(1/2),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(tanh(x) + 1)^(1/2))/2))/2 - 3/(tanh(x) + 1)^(1/2) - (2*tanh(x))/(tanh(x) + 1)^(1/2)

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sympy [A]  time = 3.58, size = 78, normalized size = 1.86 \[ - 2 \sqrt {\tanh {\relax (x )} + 1} - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {\tanh {\relax (x )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\relax (x )} + 1 > 2 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {\tanh {\relax (x )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\relax (x )} + 1 < 2 \end {cases} - \frac {1}{\sqrt {\tanh {\relax (x )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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