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

Optimal. Leaf size=11 \[ \frac{\tanh (x)}{\tanh ^2(x)+1} \]

[Out]

Tanh[x]/(1 + Tanh[x]^2)

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Rubi [A]  time = 0.0249508, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {383} \[ \frac{\tanh (x)}{\tanh ^2(x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^2 + Sinh[x]^2)^(-2),x]

[Out]

Tanh[x]/(1 + Tanh[x]^2)

Rule 383

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

Rubi steps

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

Mathematica [A]  time = 0.0027112, size = 8, normalized size = 0.73 \[ \frac{1}{2} \tanh (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^2 + Sinh[x]^2)^(-2),x]

[Out]

Tanh[2*x]/2

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Maple [B]  time = 0.023, size = 36, normalized size = 3.3 \begin{align*} -2\,{\frac{- \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+6\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2+sinh(x)^2)^2,x)

[Out]

-2*(-tanh(1/2*x)^3-tanh(1/2*x))/(tanh(1/2*x)^4+6*tanh(1/2*x)^2+1)

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Maxima [A]  time = 1.10095, size = 11, normalized size = 1. \begin{align*} \frac{1}{e^{\left (-4 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(e^(-4*x) + 1)

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Fricas [B]  time = 2.19323, size = 135, normalized size = 12.27 \begin{align*} -\frac{1}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 1)

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Sympy [B]  time = 5.28396, size = 48, normalized size = 4.36 \begin{align*} \frac{2 \tanh ^{3}{\left (\frac{x}{2} \right )}}{\tanh ^{4}{\left (\frac{x}{2} \right )} + 6 \tanh ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{2 \tanh{\left (\frac{x}{2} \right )}}{\tanh ^{4}{\left (\frac{x}{2} \right )} + 6 \tanh ^{2}{\left (\frac{x}{2} \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*tanh(x/2)**3/(tanh(x/2)**4 + 6*tanh(x/2)**2 + 1) + 2*tanh(x/2)/(tanh(x/2)**4 + 6*tanh(x/2)**2 + 1)

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Giac [A]  time = 1.09845, size = 14, normalized size = 1.27 \begin{align*} -\frac{1}{e^{\left (4 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(e^(4*x) + 1)