3.238 \(\int \cosh (x) \coth (6 x) \, dx\)

Optimal. Leaf size=38 \[ \cosh (x)-\frac {1}{6} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \tanh ^{-1}(2 \cosh (x))-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

[Out]

-1/6*arctanh(cosh(x))-1/6*arctanh(2*cosh(x))+cosh(x)-1/6*arctanh(2/3*cosh(x)*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 2073, 207} \[ \cosh (x)-\frac {1}{6} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \tanh ^{-1}(2 \cosh (x))-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Coth[6*x],x]

[Out]

-ArcTanh[Cosh[x]]/6 - ArcTanh[2*Cosh[x]]/6 - ArcTanh[(2*Cosh[x])/Sqrt[3]]/(2*Sqrt[3]) + Cosh[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 207

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

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rubi steps

\begin {align*} \int \cosh (x) \coth (6 x) \, dx &=-\operatorname {Subst}\left (\int \frac {-1+18 x^2-48 x^4+32 x^6}{2 \left (3-19 x^2+32 x^4-16 x^6\right )} \, dx,x,\cosh (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+18 x^2-48 x^4+32 x^6}{3-19 x^2+32 x^4-16 x^6} \, dx,x,\cosh (x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-2-\frac {1}{3 \left (-1+x^2\right )}-\frac {2}{-3+4 x^2}-\frac {2}{3 \left (-1+4 x^2\right )}\right ) \, dx,x,\cosh (x)\right )\right )\\ &=\cosh (x)+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (x)\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+4 x^2} \, dx,x,\cosh (x)\right )+\operatorname {Subst}\left (\int \frac {1}{-3+4 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{6} \tanh ^{-1}(\cosh (x))-\frac {1}{6} \tanh ^{-1}(2 \cosh (x))-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}}+\cosh (x)\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 95, normalized size = 2.50 \[ \frac {1}{12} \left (12 \cosh (x)-2 \sqrt {3} \tanh ^{-1}\left (\frac {2-i \tanh \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-2 \sqrt {3} \tanh ^{-1}\left (\frac {2+i \tanh \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+2 \log \left (\sinh \left (\frac {x}{2}\right )\right )-2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\log (1-2 \cosh (x))-\log (2 \cosh (x)+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Coth[6*x],x]

[Out]

(-2*Sqrt[3]*ArcTanh[(2 - I*Tanh[x/2])/Sqrt[3]] - 2*Sqrt[3]*ArcTanh[(2 + I*Tanh[x/2])/Sqrt[3]] + 12*Cosh[x] - 2
*Log[Cosh[x/2]] + Log[1 - 2*Cosh[x]] - Log[1 + 2*Cosh[x]] + 2*Log[Sinh[x/2]])/12

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fricas [B]  time = 0.57, size = 157, normalized size = 4.13 \[ \frac {6 \, \cosh \relax (x)^{2} + {\left (\sqrt {3} \cosh \relax (x) + \sqrt {3} \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 4 \, \sqrt {3} \cosh \relax (x) + 5}{2 \, \cosh \relax (x)^{2} + 2 \, \sinh \relax (x)^{2} - 1}\right ) - {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x) + 1}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x) - 1}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 12 \, \cosh \relax (x) \sinh \relax (x) + 6 \, \sinh \relax (x)^{2} + 6}{12 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(6*x),x, algorithm="fricas")

[Out]

1/12*(6*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)) - (cosh(x) + sinh(x))*log((2*cosh(x) + 1)/(cosh(x) - sinh(x))) + (cosh(x) +
sinh(x))*log((2*cosh(x) - 1)/(cosh(x) - sinh(x))) - 2*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) + 1) + 2*(cos
h(x) + sinh(x))*log(cosh(x) + sinh(x) - 1) + 12*cosh(x)*sinh(x) + 6*sinh(x)^2 + 6)/(cosh(x) + sinh(x))

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giac [B]  time = 0.12, size = 89, normalized size = 2.34 \[ \frac {1}{12} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - e^{\left (-x\right )} - e^{x}}{\sqrt {3} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 1\right ) + \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 1\right ) + \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(6*x),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.29, size = 87, normalized size = 2.29 \[ \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{6}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{6}-\frac {\ln \left (1+{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{12}+\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1-{\mathrm e}^{x}+{\mathrm e}^{2 x}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*coth(6*x),x)

[Out]

1/2*exp(x)+1/2*exp(-x)+1/6*ln(exp(x)-1)-1/6*ln(exp(x)+1)-1/12*ln(1+exp(x)+exp(2*x))+1/12*ln(1+exp(2*x)-exp(x)*
3^(1/2))*3^(1/2)-1/12*ln(1+exp(2*x)+exp(x)*3^(1/2))*3^(1/2)+1/12*ln(1-exp(x)+exp(2*x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} + \frac {1}{2} \, \int \frac {e^{\left (3 \, x\right )} - e^{x}}{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} - \frac {1}{12} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{12} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{6} \, \log \left (e^{x} + 1\right ) + \frac {1}{6} \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(6*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 101, normalized size = 2.66 \[ \frac {\ln \left (\frac {1}{3}-\frac {{\mathrm {e}}^x}{3}\right )}{6}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{3}-\frac {1}{3}\right )}{6}+\frac {{\mathrm {e}}^{-x}}{2}-\frac {\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {{\mathrm {e}}^x}{36}-\frac {1}{36}\right )}{12}+\frac {\ln \left (\frac {{\mathrm {e}}^x}{36}-\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {1}{36}\right )}{12}+\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {3}\,\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {\sqrt {3}\,{\mathrm {e}}^x}{12}-\frac {1}{12}\right )}{12}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,{\mathrm {e}}^x}{12}-\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {1}{12}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(6*x)*cosh(x),x)

[Out]

log(1/3 - exp(x)/3)/6 - log(- exp(x)/3 - 1/3)/6 + exp(-x)/2 - log(- exp(2*x)/36 - exp(x)/36 - 1/36)/12 + log(e
xp(x)/36 - exp(2*x)/36 - 1/36)/12 + exp(x)/2 - (3^(1/2)*log(- exp(2*x)/12 - (3^(1/2)*exp(x))/12 - 1/12))/12 +
(3^(1/2)*log((3^(1/2)*exp(x))/12 - exp(2*x)/12 - 1/12))/12

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \coth {\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(6*x),x)

[Out]

Integral(cosh(x)*coth(6*x), x)

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