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]

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

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [C]  time = 0.06, 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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fricas [B]  time = 0.41, size = 82, normalized size = 4.10 \[ \frac {3 \, \cosh \relax (x)^{2} + {\left (\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 4 \, \sqrt {3} \cosh \relax (x) + 5}{2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 1}\right ) + 6 \, \cosh \relax (x) \sinh \relax (x) + 3 \, \sinh \relax (x)^{2} + 3}{6 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.12, size = 45, normalized size = 2.25 \[ \frac {1}{6} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - e^{\left (-x\right )} - e^{x}}{\sqrt {3} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

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

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maple [A]  time = 0.08, size = 17, normalized size = 0.85 \[ \cosh \relax (x )-\frac {\arctanh \left (\frac {2 \cosh \relax (x ) \sqrt {3}}{3}\right ) \sqrt {3}}{3} \]

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 = 0.46, size = 153, normalized size = 7.65 \[ -\frac {1}{12} \, \sqrt {3} \log \left (\sqrt {3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (-\sqrt {3} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \arctan \left (\sqrt {3} + 2 \, e^{\left (-x\right )}\right ) + \frac {1}{6} \, \arctan \left (\sqrt {3} + 2 \, e^{x}\right ) + \frac {1}{6} \, \arctan \left (-\sqrt {3} + 2 \, e^{\left (-x\right )}\right ) + \frac {1}{6} \, \arctan \left (-\sqrt {3} + 2 \, e^{x}\right ) + \frac {1}{3} \, \arctan \left (e^{\left (-x\right )}\right ) + \frac {1}{3} \, \arctan \left (e^{x}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

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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mupad [B]  time = 1.46, size = 53, normalized size = 2.65 \[ \frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}+\frac {\sqrt {3}\,\ln \left (\frac {{\mathrm {e}}^{2\,x}}{3}-\frac {\sqrt {3}\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )}{6}-\frac {\sqrt {3}\,\ln \left (\frac {{\mathrm {e}}^{2\,x}}{3}+\frac {\sqrt {3}\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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