3.241 \(\int \cosh (x) \text {sech}(3 x) \, dx\)

Optimal. Leaf size=15 \[ \frac {\tan ^{-1}\left (\sqrt {3} \tanh (x)\right )}{\sqrt {3}} \]

[Out]

1/3*arctan(tanh(x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {203} \[ \frac {\tan ^{-1}\left (\sqrt {3} \tanh (x)\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Sech[3*x],x]

[Out]

ArcTan[Sqrt[3]*Tanh[x]]/Sqrt[3]

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}(3 x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tan ^{-1}\left (\sqrt {3} \tanh (x)\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 48, normalized size = 3.20 \[ \frac {1}{4} e^{2 x} \left (2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-e^{6 x}\right )+e^{2 x} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-e^{6 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Sech[3*x],x]

[Out]

(E^(2*x)*(2*Hypergeometric2F1[1/3, 1, 4/3, -E^(6*x)] + E^(2*x)*Hypergeometric2F1[2/3, 1, 5/3, -E^(6*x)]))/4

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fricas [B]  time = 0.47, size = 31, normalized size = 2.07 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} \cosh \relax (x) + 3 \, \sqrt {3} \sinh \relax (x)}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(3*x),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*arctan(-1/3*(sqrt(3)*cosh(x) + 3*sqrt(3)*sinh(x))/(cosh(x) - sinh(x)))

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giac [A]  time = 0.13, size = 19, normalized size = 1.27 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (2 \, x\right )} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(3*x),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(2*x) - 1))

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maple [C]  time = 0.21, size = 40, normalized size = 2.67 \[ \frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sech(3*x),x)

[Out]

1/6*I*3^(1/2)*ln(exp(2*x)-1/2+1/2*I*3^(1/2))-1/6*I*3^(1/2)*ln(exp(2*x)-1/2-1/2*I*3^(1/2))

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maxima [B]  time = 0.44, size = 114, normalized size = 7.60 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-2 \, x\right )} - 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\sqrt {3} + 2 \, e^{x}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (-\sqrt {3} + 2 \, e^{x}\right ) + \frac {1}{12} \, \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{12} \, \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{6} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{12} \, \log \left (-e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(3*x),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(-2*x) - 1)) - 1/6*sqrt(3)*arctan(sqrt(3) + 2*e^x) + 1/6*sqrt(3)*arctan(-
sqrt(3) + 2*e^x) + 1/12*log(sqrt(3)*e^x + e^(2*x) + 1) + 1/12*log(-sqrt(3)*e^x + e^(2*x) + 1) - 1/6*log(e^(2*x
) + 1) + 1/6*log(e^(-2*x) + 1) - 1/12*log(-e^(-2*x) + e^(-4*x) + 1)

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mupad [B]  time = 1.49, size = 19, normalized size = 1.27 \[ \frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,{\mathrm {e}}^{2\,x}-1\right )}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/cosh(3*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \operatorname {sech}{\left (3 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sech(3*x),x)

[Out]

Integral(cosh(x)*sech(3*x), x)

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