3.230 \(\int \cosh (x) \tanh (3 x) \, dx\)

Optimal. Leaf size=20 \[ \cosh (x)-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(ArcTanh[(2*Cosh[x])/Sqrt[3]]/Sqrt[3]) + Cosh[x]

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Rubi [A]  time = 0.0299257, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {388, 206} \[ \cosh (x)-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[(2*Cosh[x])/Sqrt[3]]/Sqrt[3]) + Cosh[x]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \cosh (x) \tanh (3 x) \, dx &=\operatorname{Subst}\left (\int \frac{1-4 x^2}{3-4 x^2} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-2 \operatorname{Subst}\left (\int \frac{1}{3-4 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{3}}\right )}{\sqrt{3}}+\cosh (x)\\ \end{align*}

Mathematica [C]  time = 0.0578828, size = 55, normalized size = 2.75 \[ \cosh (x)-\frac{\tanh ^{-1}\left (\frac{2-i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{2+i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[(2 - I*Tanh[x/2])/Sqrt[3]]/Sqrt[3]) - ArcTanh[(2 + I*Tanh[x/2])/Sqrt[3]]/Sqrt[3] + Cosh[x]

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Maple [A]  time = 0.017, size = 17, normalized size = 0.9 \begin{align*} \cosh \left ( x \right ) -{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{2\,\cosh \left ( x \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)-1/3*arctanh(2/3*cosh(x)*3^(1/2))*3^(1/2)

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Maxima [B]  time = 1.69763, size = 207, normalized size = 10.35 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (\sqrt{3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (-\sqrt{3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (-\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{6} \, \arctan \left (\sqrt{3} + 2 \, e^{\left (-x\right )}\right ) + \frac{1}{6} \, \arctan \left (\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{6} \, \arctan \left (-\sqrt{3} + 2 \, e^{\left (-x\right )}\right ) + \frac{1}{6} \, \arctan \left (-\sqrt{3} + 2 \, e^{x}\right ) + \frac{1}{3} \, \arctan \left (e^{\left (-x\right )}\right ) + \frac{1}{3} \, \arctan \left (e^{x}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(sqrt(3)*e^(-x) + e^(-2*x) + 1) + 1/12*sqrt(3)*log(-sqrt(3)*e^(-x) + e^(-2*x) + 1) - 1/12*sqr
t(3)*log(sqrt(3)*e^x + e^(2*x) + 1) + 1/12*sqrt(3)*log(-sqrt(3)*e^x + e^(2*x) + 1) + 1/6*arctan(sqrt(3) + 2*e^
(-x)) + 1/6*arctan(sqrt(3) + 2*e^x) + 1/6*arctan(-sqrt(3) + 2*e^(-x)) + 1/6*arctan(-sqrt(3) + 2*e^x) + 1/3*arc
tan(e^(-x)) + 1/3*arctan(e^x) + 1/2*e^(-x) + 1/2*e^x

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Fricas [B]  time = 2.05551, size = 275, normalized size = 13.75 \begin{align*} \frac{3 \, \cosh \left (x\right )^{2} +{\left (\sqrt{3} \cosh \left (x\right ) + \sqrt{3} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 4 \, \sqrt{3} \cosh \left (x\right ) + 5}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}\right ) + 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, \sinh \left (x\right )^{2} + 3}{6 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*cosh(x)^2 + (sqrt(3)*cosh(x) + sqrt(3)*sinh(x))*log((2*cosh(x)^2 + 2*sinh(x)^2 - 4*sqrt(3)*cosh(x) + 5)
/(2*cosh(x)^2 + 2*sinh(x)^2 - 1)) + 6*cosh(x)*sinh(x) + 3*sinh(x)^2 + 3)/(cosh(x) + sinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*tanh(3*x), x)

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Giac [B]  time = 1.15087, size = 61, normalized size = 3.05 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - e^{\left (-x\right )} - e^{x}}{\sqrt{3} + e^{\left (-x\right )} + e^{x}}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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