\(\int \frac {1}{(\text {sech}^2(x)+\tanh ^2(x))^2} \, dx\) [815]

Optimal result

Integrand size = 11, antiderivative size = 1 \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[Out]

x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4466, 8} \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[In]

Int[(Sech[x]^2 + Tanh[x]^2)^(-2),x]

[Out]

x

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4466

Int[(u_.)*((a_.) + (c_.)*sec[(d_.) + (e_.)*(x_)]^2 + (b_.)*tan[(d_.) + (e_.)*(x_)]^2)^(p_.), x_Symbol] :> Dist
[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b + c, 0]

Rubi steps \begin{align*} \text {integral}& = \int 1 \, dx \\ & = x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[In]

Integrate[(Sech[x]^2 + Tanh[x]^2)^(-2),x]

[Out]

x

Maple [A] (verified)

Time = 38.34 (sec) , antiderivative size = 2, normalized size of antiderivative = 2.00

method	result	size
risch	\(x\)	\(2\)
default	\(2 \,\operatorname {arctanh}\left (\tanh \left (\frac {x}{2}\right )\right )\)	\(8\)

[In]

int(1/(sech(x)^2+tanh(x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[In]

integrate(1/(sech(x)^2+tanh(x)^2)^2,x, algorithm="fricas")

[Out]

x

Sympy [F]

\[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=\int \frac {1}{\left (\tanh ^{2}{\left (x \right )} + \operatorname {sech}^{2}{\left (x \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(sech(x)**2+tanh(x)**2)**2,x)

[Out]

Integral((tanh(x)**2 + sech(x)**2)**(-2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[In]

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

[Out]

x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[In]

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

[Out]

x

Mupad [B] (verification not implemented)

Time = 2.19 (sec) , antiderivative size = 1, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\text {sech}^2(x)+\tanh ^2(x)\right )^2} \, dx=x \]

[In]

int(1/(1/cosh(x)^2 + tanh(x)^2)^2,x)

[Out]

x