3.113 \(\int \sqrt{1+\coth (x)} \text{sech}^2(x) \, dx\)

Optimal. Leaf size=21 \[ \tanh ^{-1}\left (\sqrt{\coth (x)+1}\right )+\tanh (x) \sqrt{\coth (x)+1} \]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]] + Sqrt[1 + Coth[x]]*Tanh[x]

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Rubi [A]  time = 0.045848, antiderivative size = 21, normalized size of antiderivative = 1., 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 = {3516, 47, 63, 207} \[ \tanh ^{-1}\left (\sqrt{\coth (x)+1}\right )+\tanh (x) \sqrt{\coth (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Coth[x]]*Sech[x]^2,x]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]] + Sqrt[1 + Coth[x]]*Tanh[x]

Rule 3516

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rubi steps

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

Mathematica [C]  time = 5.00565, size = 160, normalized size = 7.62 \[ \frac{1}{2} \sqrt{\coth (x)+1} \left (2 \tanh (x)+\frac{(1-i) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )}{\sqrt{i (\coth (x)+1)}}+\frac{\sinh \left (\frac{x}{2}\right ) \left (4 \tanh ^{-1}\left (\sqrt{\tanh \left (\frac{x}{2}\right )}\right )+\sqrt{2} \left (\log \left (\tanh \left (\frac{x}{2}\right )-\sqrt{2} \sqrt{\tanh \left (\frac{x}{2}\right )}+1\right )-\log \left (\tanh \left (\frac{x}{2}\right )+\sqrt{2} \sqrt{\tanh \left (\frac{x}{2}\right )}+1\right )\right )\right ) \left (\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )}{\sqrt{\tanh \left (\frac{x}{2}\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Coth[x]]*Sech[x]^2,x]

[Out]

(Sqrt[1 + Coth[x]]*(((1 - I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]])/Sqrt[I*(1 + Coth[x])] + ((4*ArcTanh[Sq
rt[Tanh[x/2]]] + Sqrt[2]*(Log[1 - Sqrt[2]*Sqrt[Tanh[x/2]] + Tanh[x/2]] - Log[1 + Sqrt[2]*Sqrt[Tanh[x/2]] + Tan
h[x/2]]))*Sinh[x/2]*(-Cosh[x/2] + Sinh[x/2]))/Sqrt[Tanh[x/2]] + 2*Tanh[x]))/2

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Maple [F]  time = 0.177, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (x\right ) \right ) ^{2}\sqrt{1+{\rm coth} \left (x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2*(1+coth(x))^(1/2),x)

[Out]

int(sech(x)^2*(1+coth(x))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\coth \left (x\right ) + 1} \operatorname{sech}\left (x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(coth(x) + 1)*sech(x)^2, x)

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Fricas [B]  time = 2.55175, size = 813, normalized size = 38.71 \begin{align*} \frac{4 \, \sqrt{2}{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} +{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{2 \, \sqrt{2}{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} + 3 \, \cosh \left (x\right )^{2} + 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} - 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (-\frac{2 \, \sqrt{2}{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, \cosh \left (x\right )^{2} - 6 \, \cosh \left (x\right ) \sinh \left (x\right ) - 3 \, \sinh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right )}{4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sinh(x)/(cosh(x) - sinh(x))) + (cosh(x)^2 + 2*cosh(x)*
sinh(x) + sinh(x)^2 + 1)*log((2*sqrt(2)*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sinh(x)/(cosh(x) - sinh(x)))
+ 3*cosh(x)^2 + 6*cosh(x)*sinh(x) + 3*sinh(x)^2 - 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) - (cosh(x)^2
+ 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*log(-(2*sqrt(2)*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sinh(x)/(cosh(x
) - sinh(x))) - 3*cosh(x)^2 - 6*cosh(x)*sinh(x) - 3*sinh(x)^2 + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)
))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2*(1+coth(x))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.16151, size = 201, normalized size = 9.57 \begin{align*} -\frac{1}{4} \, \sqrt{2}{\left (\sqrt{2}{\left (2 \, \sqrt{2} - \log \left (-\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right )\right )} + \sqrt{2} \log \left (\frac{{\left (\sqrt{e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} - 2 \, \sqrt{2} + 3}{{\left (\sqrt{e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 2 \, \sqrt{2} + 3}\right ) - \frac{8 \,{\left (3 \,{\left (\sqrt{e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 1\right )}}{{\left (\sqrt{e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{4} + 6 \,{\left (\sqrt{e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 1}\right )} \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(sqrt(2)*(2*sqrt(2) - log(-(sqrt(2) - 1)/(sqrt(2) + 1))) + sqrt(2)*log(((sqrt(e^(2*x) - 1) - e^x)
^2 - 2*sqrt(2) + 3)/((sqrt(e^(2*x) - 1) - e^x)^2 + 2*sqrt(2) + 3)) - 8*(3*(sqrt(e^(2*x) - 1) - e^x)^2 + 1)/((s
qrt(e^(2*x) - 1) - e^x)^4 + 6*(sqrt(e^(2*x) - 1) - e^x)^2 + 1))*sgn(e^(2*x) - 1)