3.234 \(\int \cosh (x) \coth (2 x) \, dx\)

Optimal. Leaf size=10 \[ \cosh (x)-\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {12, 388, 206} \[ \cosh (x)-\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Coth[2*x],x]

[Out]

-ArcTanh[Cosh[x]]/2 + 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 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])

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rubi steps

\begin {align*} \int \cosh (x) \coth (2 x) \, dx &=-\operatorname {Subst}\left (\int \frac {-1+2 x^2}{2 \left (1-x^2\right )} \, dx,x,\cosh (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+2 x^2}{1-x^2} \, dx,x,\cosh (x)\right )\right )\\ &=\cosh (x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))+\cosh (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.40 \[ \cosh (x)+\frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Coth[2*x],x]

[Out]

Cosh[x] + Log[Tanh[x/2]]/2

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fricas [B]  time = 0.41, size = 52, normalized size = 5.20 \[ \frac {\cosh \relax (x)^{2} - {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(2*x),x, algorithm="fricas")

[Out]

1/2*(cosh(x)^2 - (cosh(x) + sinh(x))*log(cosh(x) + sinh(x) + 1) + (cosh(x) + sinh(x))*log(cosh(x) + sinh(x) -
1) + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)/(cosh(x) + sinh(x))

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giac [B]  time = 0.11, size = 26, normalized size = 2.60 \[ \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(2*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.15, size = 9, normalized size = 0.90 \[ \cosh \relax (x )-\arctanh \left ({\mathrm e}^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*coth(2*x),x)

[Out]

cosh(x)-arctanh(exp(x))

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maxima [B]  time = 0.36, size = 29, normalized size = 2.90 \[ \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(2*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.41, size = 29, normalized size = 2.90 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(2*x)*cosh(x),x)

[Out]

log(1 - exp(x))/2 - log(- exp(x) - 1)/2 + exp(-x)/2 + exp(x)/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \coth {\left (2 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(2*x),x)

[Out]

Integral(cosh(x)*coth(2*x), x)

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