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

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Rubi [A]  time = 0.05, antiderivative size = 21, 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 = {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 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])

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]

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*}

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Mathematica [C]  time = 5.27, 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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fricas [B]  time = 0.43, size = 231, normalized size = 11.00 \[ \frac {4 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} + {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {2 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} + 3 \, \cosh \relax (x)^{2} + 6 \, \cosh \relax (x) \sinh \relax (x) + 3 \, \sinh \relax (x)^{2} - 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (-\frac {2 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 3 \, \cosh \relax (x)^{2} - 6 \, \cosh \relax (x) \sinh \relax (x) - 3 \, \sinh \relax (x)^{2} + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right )}{4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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giac [B]  time = 0.13, size = 149, normalized size = 7.10 \[ -\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} {\left (2 \, \sqrt {2} - \log \left (-\frac {\sqrt {2} - 1}{\sqrt {2} + 1}\right )\right )} + \sqrt {2} \log \left (\frac {{\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} - 2 \, \sqrt {2} + 3}{{\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 2 \, \sqrt {2} + 3}\right ) - \frac {8 \, {\left (3 \, {\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 1\right )}}{{\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{4} + 6 \, {\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 1}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]

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)

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maple [F]  time = 0.38, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\relax (x )^{2} \sqrt {1+\coth \relax (x )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\coth \relax (x) + 1} \operatorname {sech}\relax (x)^{2}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {\mathrm {coth}\relax (x)+1}}{{\mathrm {cosh}\relax (x)}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x) + 1)^(1/2)/cosh(x)^2,x)

[Out]

int((coth(x) + 1)^(1/2)/cosh(x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\coth {\relax (x )} + 1} \operatorname {sech}^{2}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(coth(x) + 1)*sech(x)**2, x)

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