### 3.1001 $$\int \sqrt{1+\coth ^2(x)} \text{sech}^2(x) \, dx$$

Optimal. Leaf size=19 $\tanh (x) \sqrt{\coth ^2(x)+1}-\sinh ^{-1}(\coth (x))$

[Out]

-ArcSinh[Coth[x]] + Sqrt[1 + Coth[x]^2]*Tanh[x]

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Rubi [A]  time = 0.0491916, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {3663, 277, 215} $\tanh (x) \sqrt{\coth ^2(x)+1}-\sinh ^{-1}(\coth (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[1 + Coth[x]^2]*Sech[x]^2,x]

[Out]

-ArcSinh[Coth[x]] + Sqrt[1 + Coth[x]^2]*Tanh[x]

Rule 3663

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [B]  time = 0.208612, size = 51, normalized size = 2.68 $\sinh (x) \sqrt{\coth ^2(x)+1} \text{sech}(2 x) \left (\cosh (x)+\sinh (x) \tanh (x)-\sqrt{-\cosh (2 x)} \tan ^{-1}\left (\frac{\cosh (x)}{\sqrt{-\cosh (2 x)}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sqrt[1 + Coth[x]^2]*Sech[2*x]*Sinh[x]*(Cosh[x] - ArcTan[Cosh[x]/Sqrt[-Cosh[2*x]]]*Sqrt[-Cosh[2*x]] + Sinh[x]*T
anh[x])

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Maple [F]  time = 0.232, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (x\right ) \right ) ^{2}\sqrt{1+ \left ({\rm coth} \left (x\right ) \right ) ^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\coth \left (x\right )^{2} + 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)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.06866, size = 786, normalized size = 41.37 \begin{align*} -\frac{{\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{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \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{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \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 \, \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\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)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/2*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*log((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 2*sqrt(
(cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sin
h(x)^2)) - (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*log((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 2*
sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x)
+ sinh(x)^2)) - 4*sqrt((cosh(x)^2 + sinh(x)^2)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^2 + 2*co
sh(x)*sinh(x) + sinh(x)^2 + 1)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17636, size = 162, normalized size = 8.53 \begin{align*} \frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \,{\left (\sqrt{2} + \sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) - \frac{4 \,{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 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)^2)^(1/2),x, algorithm="giac")

[Out]

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