3.240 \(\int \cosh (x) \text{sech}(2 x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]

[Out]

ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]

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Rubi [A]  time = 0.0164232, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4356, 203} \[ \frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Sech[2*x],x]

[Out]

ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]

Rule 4356

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0079444, size = 15, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Sech[2*x],x]

[Out]

ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]

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Maple [C]  time = 0.033, size = 44, normalized size = 2.9 \begin{align*}{\frac{i}{4}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{2}{{\rm e}^{x}}-1 \right ) -{\frac{i}{4}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{2}{{\rm e}^{x}}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sech(2*x),x)

[Out]

1/4*I*2^(1/2)*ln(exp(2*x)+I*2^(1/2)*exp(x)-1)-1/4*I*2^(1/2)*ln(exp(2*x)-I*2^(1/2)*exp(x)-1)

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Maxima [B]  time = 1.76938, size = 58, normalized size = 3.87 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^(-x))) - 1/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^(-x)))

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Fricas [B]  time = 2.02641, size = 254, normalized size = 16.93 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \cosh \left (x\right ) + \frac{1}{2} \, \sqrt{2} \sinh \left (x\right )\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} + \sqrt{2}}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*cosh(x) + 1/2*sqrt(2)*sinh(x)) - 1/2*sqrt(2)*arctan(-1/2*(sqrt(2)*cosh(x)^2 + 2
*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))/(cosh(x) - sinh(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \operatorname{sech}{\left (2 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*sech(2*x), x)

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Giac [B]  time = 1.12215, size = 53, normalized size = 3.53 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) + 1/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x))