3.996 \(\int (1+\cosh ^2(x)) \text {sech}^2(x) \, dx\)

Optimal. Leaf size=4 \[ x+\tanh (x) \]

[Out]

x+tanh(x)

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Rubi [A]  time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3012, 8} \[ x+\tanh (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^2)*Sech[x]^2,x]

[Out]

x + Tanh[x]

Rule 8

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

Rule 3012

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \left (1+\cosh ^2(x)\right ) \text {sech}^2(x) \, dx &=\tanh (x)+\int 1 \, dx\\ &=x+\tanh (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 4, normalized size = 1.00 \[ x+\tanh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^2)*Sech[x]^2,x]

[Out]

x + Tanh[x]

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fricas [B]  time = 0.47, size = 14, normalized size = 3.50 \[ \frac {{\left (x - 1\right )} \cosh \relax (x) + \sinh \relax (x)}{\cosh \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cosh(x)^2)*sech(x)^2,x, algorithm="fricas")

[Out]

((x - 1)*cosh(x) + sinh(x))/cosh(x)

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giac [B]  time = 0.11, size = 12, normalized size = 3.00 \[ x - \frac {2}{e^{\left (2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2/(e^(2*x) + 1)

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maple [A]  time = 0.37, size = 5, normalized size = 1.25 \[ x +\tanh \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cosh(x)^2)*sech(x)^2,x)

[Out]

x+tanh(x)

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maxima [B]  time = 0.31, size = 12, normalized size = 3.00 \[ x + \frac {2}{e^{\left (-2 \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2/(e^(-2*x) + 1)

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mupad [B]  time = 1.65, size = 12, normalized size = 3.00 \[ x-\frac {2}{{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^2 + 1)/cosh(x)^2,x)

[Out]

x - 2/(exp(2*x) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\cosh ^{2}{\relax (x )} + 1\right ) \operatorname {sech}^{2}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cosh(x)**2 + 1)*sech(x)**2, x)

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