3.36 \(\int \frac{1}{(1+\cosh ^2(x))^2} \, dx\)

Optimal. Leaf size=35 \[ \frac{3 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{\sinh (x) \cosh (x)}{4 \left (\cosh ^2(x)+1\right )} \]

[Out]

(3*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - (Cosh[x]*Sinh[x])/(4*(1 + Cosh[x]^2))

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Rubi [A]  time = 0.0252306, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3184, 12, 3181, 206} \[ \frac{3 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{\sinh (x) \cosh (x)}{4 \left (\cosh ^2(x)+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^2)^(-2),x]

[Out]

(3*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - (Cosh[x]*Sinh[x])/(4*(1 + Cosh[x]^2))

Rule 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
e + f*x]^2)^(p + 1))/(2*a*f*(p + 1)*(a + b)), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^(p
+ 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ
[a + b, 0] && LtQ[p, -1]

Rule 12

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

Rule 3181

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

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

Mathematica [A]  time = 0.127925, size = 35, normalized size = 1. \[ \frac{3 \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{\sinh (2 x)}{4 (\cosh (2 x)+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^2)^(-2),x]

[Out]

(3*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - Sinh[2*x]/(4*(3 + Cosh[2*x]))

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Maple [B]  time = 0.014, size = 113, normalized size = 3.2 \begin{align*} -{\frac{1}{2} \left ({\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) } \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}+1 \right ) ^{-1}}+{\frac{3\,\sqrt{2}}{32}\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{3\,\sqrt{2}}{32}\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/(1+cosh(x)^2)^2,x)

[Out]

-1/2*(1/2*tanh(1/2*x)^3+1/2*tanh(1/2*x))/(tanh(1/2*x)^4+1)+3/32*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))-3/32*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 = 1.70208, size = 80, normalized size = 2.29 \begin{align*} -\frac{3}{16} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac{3 \, e^{\left (-2 \, x\right )} + 1}{2 \,{\left (6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*sqrt(2)*log(-(2*sqrt(2) - e^(-2*x) - 3)/(2*sqrt(2) + e^(-2*x) + 3)) - 1/2*(3*e^(-2*x) + 1)/(6*e^(-2*x) +
e^(-4*x) + 1)

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Fricas [B]  time = 2.12037, size = 728, normalized size = 20.8 \begin{align*} \frac{24 \, \cosh \left (x\right )^{2} + 3 \,{\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 ) + 48 \, \cosh \left (x\right ) \sinh \left (x\right ) + 24 \, \sinh \left (x\right )^{2} + 8}{16 \,{\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/(1+cosh(x)^2)^2,x, algorithm="fricas")

[Out]

1/16*(24*cosh(x)^2 + 3*(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)/(cosh(x)^2 + sinh(x)^2 + 3)) + 48*cosh(x)*sinh(x) + 24*sinh(x)^2 + 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 [B]  time = 6.58456, size = 211, normalized size = 6.03 \begin{align*} - \frac{3 \sqrt{2} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} - 4 \sqrt{2} \tanh{\left (\frac{x}{2} \right )} + 4 \right )} \tanh ^{4}{\left (\frac{x}{2} \right )}}{16 \tanh ^{4}{\left (\frac{x}{2} \right )} + 16} - \frac{3 \sqrt{2} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} - 4 \sqrt{2} \tanh{\left (\frac{x}{2} \right )} + 4 \right )}}{16 \tanh ^{4}{\left (\frac{x}{2} \right )} + 16} + \frac{3 \sqrt{2} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} + 4 \sqrt{2} \tanh{\left (\frac{x}{2} \right )} + 4 \right )} \tanh ^{4}{\left (\frac{x}{2} \right )}}{16 \tanh ^{4}{\left (\frac{x}{2} \right )} + 16} + \frac{3 \sqrt{2} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} + 4 \sqrt{2} \tanh{\left (\frac{x}{2} \right )} + 4 \right )}}{16 \tanh ^{4}{\left (\frac{x}{2} \right )} + 16} - \frac{4 \tanh ^{3}{\left (\frac{x}{2} \right )}}{16 \tanh ^{4}{\left (\frac{x}{2} \right )} + 16} - \frac{4 \tanh{\left (\frac{x}{2} \right )}}{16 \tanh ^{4}{\left (\frac{x}{2} \right )} + 16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)**2)**2,x)

[Out]

-3*sqrt(2)*log(4*tanh(x/2)**2 - 4*sqrt(2)*tanh(x/2) + 4)*tanh(x/2)**4/(16*tanh(x/2)**4 + 16) - 3*sqrt(2)*log(4
*tanh(x/2)**2 - 4*sqrt(2)*tanh(x/2) + 4)/(16*tanh(x/2)**4 + 16) + 3*sqrt(2)*log(4*tanh(x/2)**2 + 4*sqrt(2)*tan
h(x/2) + 4)*tanh(x/2)**4/(16*tanh(x/2)**4 + 16) + 3*sqrt(2)*log(4*tanh(x/2)**2 + 4*sqrt(2)*tanh(x/2) + 4)/(16*
tanh(x/2)**4 + 16) - 4*tanh(x/2)**3/(16*tanh(x/2)**4 + 16) - 4*tanh(x/2)/(16*tanh(x/2)**4 + 16)

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Giac [B]  time = 1.35931, size = 80, normalized size = 2.29 \begin{align*} \frac{3}{16} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) + \frac{3 \, e^{\left (2 \, x\right )} + 1}{2 \,{\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*sqrt(2)*log(-(2*sqrt(2) - e^(2*x) - 3)/(2*sqrt(2) + e^(2*x) + 3)) + 1/2*(3*e^(2*x) + 1)/(e^(4*x) + 6*e^(2
*x) + 1)