3.126 \(\int \tanh (x) \sqrt {1+\tanh (x)} \, dx\)

Optimal. Leaf size=32 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\tanh (x)+1} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 32, 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 = {3527, 3480, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\tanh (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]*Sqrt[1 + Tanh[x]],x]

[Out]

Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 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 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 3527

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 32, normalized size = 1.00 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\tanh (x)+1} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]*Sqrt[1 + Tanh[x]],x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 129, normalized size = 4.03 \[ -\frac {4 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - {\left (\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} - 2 \, \cosh \relax (x)^{2} - 4 \, \cosh \relax (x) \sinh \relax (x) - 2 \, \sinh \relax (x)^{2} - 1\right )}{2 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.15, size = 53, normalized size = 1.66 \[ \frac {1}{2} \, \sqrt {2} {\left (\frac {4}{\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1} - \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 26, normalized size = 0.81 \[ \arctanh \left (\frac {\sqrt {1+\tanh \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\sqrt {2}}{\sqrt {e^{\left (-2 \, x\right )} + 1}} + \int \frac {\sqrt {2} e^{\left (-x\right )}}{{\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )} \sqrt {e^{\left (-2 \, x\right )} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.06, size = 25, normalized size = 0.78 \[ \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\relax (x)+1}}{2}\right )-2\,\sqrt {\mathrm {tanh}\relax (x)+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)*(tanh(x) + 1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 2.12, size = 70, normalized size = 2.19 \[ - 2 \sqrt {\tanh {\relax (x )} + 1} - 2 \left (\begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {\tanh {\relax (x )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\relax (x )} + 1 > 2 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {\tanh {\relax (x )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\relax (x )} + 1 < 2 \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________