3.247 \(\int \cosh (x) \text{csch}(4 x) \, dx\)

Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cosh (x)) \]

[Out]

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

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Rubi [A]  time = 0.03261, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1093, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \cosh (x)\right )}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1093

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

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])

Rubi steps

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

Mathematica [C]  time = 0.294351, size = 183, normalized size = 7.04 \[ \frac{4 \tanh ^{-1}\left (\sqrt{2}-i \tanh \left (\frac{x}{2}\right )\right )+2 \sqrt{2} \log \left (\sinh \left (\frac{x}{2}\right )\right )-2 \sqrt{2} \log \left (\cosh \left (\frac{x}{2}\right )\right )-\log \left (\sqrt{2}-2 \cosh (x)\right )+\log \left (2 \cosh (x)+\sqrt{2}\right )+2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cosh \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sinh \left (\frac{x}{2}\right )}\right )-2 i \tan ^{-1}\left (\frac{\sinh \left (\frac{x}{2}\right )+\cosh \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cosh \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sinh \left (\frac{x}{2}\right )}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Csch[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[(Co
sh[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]]
- 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 = 0.039, size = 53, normalized size = 2. \begin{align*} -{\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)*csch(4*x),x)

[Out]

-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+exp(2*x)-exp(x)*2^(1/2))
*2^(1/2)

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Maxima [B]  time = 1.68406, size = 81, normalized size = 3.12 \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}{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)*csch(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) - 1/4*log(e^(
-x) + 1) + 1/4*log(e^(-x) - 1)

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Fricas [B]  time = 2.10151, size = 211, normalized size = 8.12 \begin{align*} \frac{1}{8} \, \sqrt{2} \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 ) - \frac{1}{4} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(4*x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*csch(4*x), x)

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Giac [B]  time = 1.14539, size = 77, normalized size = 2.96 \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}{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)*csch(4*x),x, algorithm="giac")

[Out]

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