3.9.9 \(\int \frac {1}{(\cosh ^2(x)+\sinh ^2(x))^2} \, dx\) [809]

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

[Out]

tanh(x)/(1+tanh(x)^2)

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Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {391} \begin {gather*} \frac {\tanh (x)}{\tanh ^2(x)+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^2 + Sinh[x]^2)^(-2),x]

[Out]

Tanh[x]/(1 + Tanh[x]^2)

Rule 391

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 &=\text {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*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 0.73 \begin {gather*} \frac {1}{2} \tanh (2 x) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(11)=22\).
time = 1.11, size = 36, normalized size = 3.27

method	result	size
risch	\(-\frac {1}{1+{\mathrm e}^{4 x}}\)	\(11\)
default	\(-\frac {2 \left (-\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-\tanh \left (\frac {x}{2}\right )\right )}{\tanh ^{4}\left (\frac {x}{2}\right )+6 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+1}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.26, size = 8, normalized size = 0.73 \begin {gather*} \frac {1}{e^{\left (-4 \, x\right )} + 1} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).
time = 0.37, size = 40, normalized size = 3.64 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (8) = 16\).
time = 0.67, size = 48, normalized size = 4.36 \begin {gather*} \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 {gather*}

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 = 0.41, size = 10, normalized size = 0.91 \begin {gather*} -\frac {1}{e^{\left (4 \, x\right )} + 1} \end {gather*}

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)

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Mupad [B]
time = 1.55, size = 10, normalized size = 0.91 \begin {gather*} -\frac {1}{{\mathrm {e}}^{4\,x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2 + sinh(x)^2)^2,x)

[Out]

-1/(exp(4*x) + 1)

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