### 3.820 $$\int \frac{1}{\coth ^2(x)+\text{csch}^2(x)} \, dx$$

Optimal. Leaf size=18 $x-\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )$

[Out]

x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

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Rubi [A]  time = 0.0302392, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {1130, 207} $x-\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x]^2 + Csch[x]^2)^(-1),x]

[Out]

x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

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 \frac{1}{\coth ^2(x)+\text{csch}^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{x^2}{2-3 x^2+x^4} \, dx,x,\tanh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{-2+x^2} \, dx,x,\tanh (x)\right )-\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (x)\right )\\ &=x-\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0806825, size = 18, normalized size = 1. $x-\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x]^2 + Csch[x]^2)^(-1),x]

[Out]

x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

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Maple [B]  time = 0.034, size = 102, normalized size = 5.7 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -{\frac{\sqrt{2}}{4}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)^2+csch(x)^2),x)

[Out]

ln(tanh(1/2*x)+1)-ln(tanh(1/2*x)-1)-1/4*2^(1/2)*ln((tanh(1/2*x)^2+2^(1/2)*tanh(1/2*x)+1)/(tanh(1/2*x)^2-2^(1/2
)*tanh(1/2*x)+1))+1/4*2^(1/2)*ln((tanh(1/2*x)^2-2^(1/2)*tanh(1/2*x)+1)/(tanh(1/2*x)^2+2^(1/2)*tanh(1/2*x)+1))

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Maxima [B]  time = 1.69103, size = 49, normalized size = 2.72 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, x\right )} + 3}\right ) + x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.27656, size = 219, normalized size = 12.17 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\frac{3 \,{\left (2 \, \sqrt{2} + 3\right )} \cosh \left (x\right )^{2} - 4 \,{\left (3 \, \sqrt{2} + 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (2 \, \sqrt{2} + 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) + x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(coth(x)**2 + csch(x)**2), x)

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Giac [B]  time = 1.13117, size = 49, normalized size = 2.72 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) + x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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