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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0859348, antiderivative size = 38, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [C]  time = 0.0711355, size = 95, normalized size = 2.5 \[ \frac{1}{12} \left (12 \cosh (x)-2 \sqrt{3} \tanh ^{-1}\left (\frac{2-i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )-2 \sqrt{3} \tanh ^{-1}\left (\frac{2+i \tanh \left (\frac{x}{2}\right )}{\sqrt{3}}\right )+2 \log \left (\sinh \left (\frac{x}{2}\right )\right )-2 \log \left (\cosh \left (\frac{x}{2}\right )\right )+\log (1-2 \cosh (x))-\log (2 \cosh (x)+1)\right ) \]

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

________________________________________________________________________________________

Maple [B]  time = 0.097, size = 87, normalized size = 2.3 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}+{\frac{{{\rm e}^{-x}}}{2}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{6}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{6}}+{\frac{\ln \left ({{\rm e}^{2\,x}}-{{\rm e}^{x}}+1 \right ) }{12}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+{{\rm e}^{x}}+1 \right ) }{12}}+{\frac{\sqrt{3}\ln \left ({{\rm e}^{2\,x}}-\sqrt{3}{{\rm e}^{x}}+1 \right ) }{12}}-{\frac{\sqrt{3}\ln \left ({{\rm e}^{2\,x}}+\sqrt{3}{{\rm e}^{x}}+1 \right ) }{12}} \end{align*}

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 2.12742, size = 586, normalized size = 15.42 \begin{align*} \frac{6 \, \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 ) -{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right ) - 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 12 \, \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, \sinh \left (x\right )^{2} + 6}{12 \,{\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)*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))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \coth{\left (6 x \right )}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [B]  time = 1.13602, size = 120, normalized size = 3.16 \begin{align*} \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 ) \end{align*}

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)