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]

1/2*arctan(sinh(x)*2^(1/2))*2^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 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])

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])

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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 68, normalized size = 4.53 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \cosh \relax (x) + \frac {1}{2} \, \sqrt {2} \sinh \relax (x)\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}}{2 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) \]

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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giac [B]  time = 0.13, size = 39, normalized size = 2.60 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) \]

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))

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maple [C]  time = 0.22, size = 44, normalized size = 2.93 \[ \frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{4}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{4} \]

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 = 0.44, size = 43, normalized size = 2.87 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) \]

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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mupad [B]  time = 0.09, size = 32, normalized size = 2.13 \[ \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {\sqrt {2}\,{\mathrm {e}}^{3\,x}}{2}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{2}\right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*(atan((2^(1/2)*exp(x))/2 + (2^(1/2)*exp(3*x))/2) + atan((2^(1/2)*exp(x))/2)))/2

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

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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