### 3.821 $$\int \frac{1}{(\coth ^2(x)+\text{csch}^2(x))^2} \, dx$$

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

[Out]

x - ArcTanh[Tanh[x]/Sqrt[2]]/Sqrt[2] - Tanh[x]/(2 - Tanh[x]^2)

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Rubi [A]  time = 0.0467893, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {470, 12, 391, 206} $x-\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh (x)}{2-\tanh ^2(x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x]^2 + Csch[x]^2)^(-2),x]

[Out]

x - ArcTanh[Tanh[x]/Sqrt[2]]/Sqrt[2] - Tanh[x]/(2 - Tanh[x]^2)

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

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])

Rubi steps

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

Mathematica [A]  time = 0.130939, size = 64, normalized size = 2. $\frac{(\cosh (2 x)+3) \text{csch}^4(x) \left (6 x-2 \sinh (2 x)+2 x \cosh (2 x)-\sqrt{2} (\cosh (2 x)+3) \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )\right )}{8 \left (\coth ^2(x)+\text{csch}^2(x)\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((3 + Cosh[2*x])*Csch[x]^4*(6*x + 2*x*Cosh[2*x] - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]*(3 + Cosh[2*x]) - 2*Sinh[2*
x]))/(8*(Coth[x]^2 + Csch[x]^2)^2)

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Maple [B]  time = 0.048, size = 129, normalized size = 4. \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,{\frac{-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/2\,\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+1}}-{\frac{\sqrt{2}}{8}\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}}{8}\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)^2,x)

[Out]

ln(tanh(1/2*x)+1)-ln(tanh(1/2*x)-1)+2*(-1/2*tanh(1/2*x)^3-1/2*tanh(1/2*x))/(tanh(1/2*x)^4+1)-1/8*2^(1/2)*ln((t
anh(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))+1/8*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 = 1.69692, size = 81, normalized size = 2.53 \begin{align*} \frac{1}{4} \, \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 - \frac{2 \,{\left (3 \, e^{\left (-2 \, x\right )} + 1\right )}}{6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.35679, size = 883, normalized size = 27.59 \begin{align*} \frac{4 \, x \cosh \left (x\right )^{4} + 16 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 4 \, x \sinh \left (x\right )^{4} + 24 \,{\left (x + 1\right )} \cosh \left (x\right )^{2} + 24 \,{\left (x \cosh \left (x\right )^{2} + x + 1\right )} \sinh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right )^{4} + 4 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt{2} \sinh \left (x\right )^{4} + 6 \,{\left (\sqrt{2} \cosh \left (x\right )^{2} + \sqrt{2}\right )} \sinh \left (x\right )^{2} + 6 \, \sqrt{2} \cosh \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cosh \left (x\right )^{3} + 3 \, \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt{2}\right )} \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 ) + 16 \,{\left (x \cosh \left (x\right )^{3} + 3 \,{\left (x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, x + 8}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \,{\left (\cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x*cosh(x)^4 + 16*x*cosh(x)*sinh(x)^3 + 4*x*sinh(x)^4 + 24*(x + 1)*cosh(x)^2 + 24*(x*cosh(x)^2 + x + 1)*
sinh(x)^2 + (sqrt(2)*cosh(x)^4 + 4*sqrt(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)^4 + 6*(sqrt(2)*cosh(x)^2 + sqrt
(2))*sinh(x)^2 + 6*sqrt(2)*cosh(x)^2 + 4*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x) + 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)/(cos
h(x)^2 + sinh(x)^2 + 3)) + 16*(x*cosh(x)^3 + 3*(x + 1)*cosh(x))*sinh(x) + 4*x + 8)/(cosh(x)^4 + 4*cosh(x)*sinh
(x)^3 + sinh(x)^4 + 6*(cosh(x)^2 + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 + 3*cosh(x))*sinh(x) + 1)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x)**2 + csch(x)**2)**(-2), x)

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Giac [B]  time = 1.15796, size = 81, normalized size = 2.53 \begin{align*} -\frac{1}{4} \, \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 + \frac{2 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )}}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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