### 3.236 $$\int \cosh (x) \coth (4 x) \, dx$$

Optimal. Leaf size=28 $\cosh (x)-\frac{1}{4} \tanh ^{-1}(\cosh (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}$

[Out]

-ArcTanh[Cosh[x]]/4 - ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2]) + Cosh[x]

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Rubi [A]  time = 0.0567339, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {1676, 1166, 207} $\cosh (x)-\frac{1}{4} \tanh ^{-1}(\cosh (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[x]*Coth[4*x],x]

[Out]

-ArcTanh[Cosh[x]]/4 - ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2]) + Cosh[x]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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 (4 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+8 x^2-8 x^4}{4-12 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1+\frac{3-4 x^2}{4-12 x^2+8 x^4}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\operatorname{Subst}\left (\int \frac{3-4 x^2}{4-12 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+2 \operatorname{Subst}\left (\int \frac{1}{-8+8 x^2} \, dx,x,\cosh (x)\right )+2 \operatorname{Subst}\left (\int \frac{1}{-4+8 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{4} \tanh ^{-1}(\cosh (x))-\frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}+\cosh (x)\\ \end{align*}

Mathematica [C]  time = 0.224495, size = 192, normalized size = 6.86 $\frac{8 \sqrt{2} \cosh (x)-4 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+2 \sqrt{2} \log \left (\sinh \left (\frac{x}{2}\right )\right )-2 \sqrt{2} \log \left (\cosh \left (\frac{x}{2}\right )\right )+\log \left (\sqrt{2}-2 \cosh (x)\right )-\log \left (2 \cosh (x)+\sqrt{2}\right )-2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )+2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{8 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[x]*Coth[4*x],x]

[Out]

((-2*I)*ArcTan[(Cosh[x/2] + Sinh[x/2])/((1 + Sqrt[2])*Cosh[x/2] - (-1 + Sqrt[2])*Sinh[x/2])] + (2*I)*ArcTan[(C
osh[x/2] + Sinh[x/2])/((-1 + Sqrt[2])*Cosh[x/2] - (1 + Sqrt[2])*Sinh[x/2])] - 4*ArcTanh[Sqrt[2] - I*Tanh[x/2]]
+ 8*Sqrt[2]*Cosh[x] - 2*Sqrt[2]*Log[Cosh[x/2]] + Log[Sqrt[2] - 2*Cosh[x]] - Log[Sqrt[2] + 2*Cosh[x]] + 2*Sqrt
[2]*Log[Sinh[x/2]])/(8*Sqrt[2])

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Maple [B]  time = 0.073, size = 63, normalized size = 2.3 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}+{\frac{{{\rm e}^{-x}}}{2}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{4}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{4}}+{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*coth(4*x),x)

[Out]

1/2*exp(x)+1/2*exp(-x)-1/4*ln(exp(x)+1)+1/4*ln(exp(x)-1)+1/8*ln(1+exp(2*x)-exp(x)*2^(1/2))*2^(1/2)-1/8*ln(1+ex
p(2*x)+exp(x)*2^(1/2))*2^(1/2)

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Maxima [B]  time = 1.66876, size = 95, normalized size = 3.39 \begin{align*} -\frac{1}{8} \, \sqrt{2} \log \left (\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{8} \, \sqrt{2} \log \left (-\sqrt{2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \frac{1}{4} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{4} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(4*x),x, algorithm="maxima")

[Out]

-1/8*sqrt(2)*log(sqrt(2)*e^(-x) + e^(-2*x) + 1) + 1/8*sqrt(2)*log(-sqrt(2)*e^(-x) + e^(-2*x) + 1) + 1/2*e^(-x)
+ 1/2*e^x - 1/4*log(e^(-x) + 1) + 1/4*log(e^(-x) - 1)

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Fricas [B]  time = 2.20536, size = 397, normalized size = 14.18 \begin{align*} \frac{4 \, \cosh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\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 ) + 8 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} + 4}{8 \,{\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(4*x),x, algorithm="fricas")

[Out]

1/8*(4*cosh(x)^2 + (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*log((cosh(x)^2 + sinh(x)^2 - 2*sqrt(2)*cosh(x) + 2)/(co
sh(x)^2 + sinh(x)^2)) - 2*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) + 1) + 2*(cosh(x) + sinh(x))*log(cosh(x) +
sinh(x) - 1) + 8*cosh(x)*sinh(x) + 4*sinh(x)^2 + 4)/(cosh(x) + sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \coth{\left (4 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(4*x),x)

[Out]

Integral(cosh(x)*coth(4*x), x)

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Giac [B]  time = 1.16048, size = 90, normalized size = 3.21 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - e^{x}}{\sqrt{2} + e^{\left (-x\right )} + e^{x}}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \frac{1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac{1}{8} \, \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(4*x),x, algorithm="giac")

[Out]

1/8*sqrt(2)*log(-(sqrt(2) - e^(-x) - e^x)/(sqrt(2) + e^(-x) + e^x)) + 1/2*e^(-x) + 1/2*e^x - 1/8*log(e^(-x) +
e^x + 2) + 1/8*log(e^(-x) + e^x - 2)