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]

-1/4*arctanh(cosh(x))+cosh(x)-1/4*arctanh(cosh(x)*2^(1/2))*2^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 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 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 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

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*}

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Mathematica [C]  time = 0.24, 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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fricas [B]  time = 0.49, size = 101, normalized size = 3.61 \[ \frac {4 \, \cosh \relax (x)^{2} + {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \log \left (\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 2 \, \sqrt {2} \cosh \relax (x) + 2}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}\right ) - 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 8 \, \cosh \relax (x) \sinh \relax (x) + 4 \, \sinh \relax (x)^{2} + 4}{8 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

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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giac [B]  time = 0.12, size = 67, normalized size = 2.39 \[ \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - e^{x}}{\sqrt {2} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {1}{8} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]

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)

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maple [B]  time = 0.26, size = 63, normalized size = 2.25 \[ \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}+\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{8}-\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{8} \]

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 = 0.44, size = 70, normalized size = 2.50 \[ -\frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{4} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{4} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

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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mupad [B]  time = 1.45, size = 71, normalized size = 2.54 \[ \frac {\ln \left (\frac {1}{2}-\frac {{\mathrm {e}}^x}{2}\right )}{4}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{2}-\frac {1}{2}\right )}{4}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {2}\,\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {\sqrt {2}\,{\mathrm {e}}^x}{8}-\frac {1}{8}\right )}{8}+\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {1}{8}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \coth {\left (4 x \right )}\, dx \]

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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