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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0317943, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [C]  time = 0.0218634, size = 48, normalized size = 3.2 \[ \frac{1}{4} e^{2 x} \left (2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-e^{6 x}\right )+e^{2 x} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-e^{6 x}\right )\right ) \]

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

________________________________________________________________________________________

Maple [C]  time = 0.035, size = 40, normalized size = 2.7 \begin{align*}{\frac{i}{6}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) -{\frac{i}{6}}\sqrt{3}\ln \left ({{\rm e}^{2\,x}}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \end{align*}

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))

________________________________________________________________________________________

Maxima [B]  time = 1.76901, size = 154, normalized size = 10.27 \begin{align*} -\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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 2.06588, size = 115, normalized size = 7.67 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} \cosh \left (x\right ) + 3 \, \sqrt{3} \sinh \left (x\right )}{3 \,{\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(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)))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \operatorname{sech}{\left (3 x \right )}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.13974, size = 26, normalized size = 1.73 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, e^{\left (2 \, x\right )} - 1\right )}\right ) \end{align*}

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))