3.216 \(\int e^x \coth (2 x) \, dx\)

Optimal. Leaf size=16 \[ e^x-\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

[Out]

E^x - ArcTan[E^x] - ArcTanh[E^x]

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Rubi [A]  time = 0.0141921, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {2282, 388, 212, 206, 203} \[ e^x-\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Coth[2*x],x]

[Out]

E^x - ArcTan[E^x] - ArcTanh[E^x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

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]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int e^x \coth (2 x) \, dx &=\operatorname{Subst}\left (\int \frac{-1-x^4}{1-x^4} \, dx,x,e^x\right )\\ &=e^x-2 \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,e^x\right )\\ &=e^x-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )\\ &=e^x-\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0119652, size = 16, normalized size = 1. \[ e^x-\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Coth[2*x],x]

[Out]

E^x - ArcTan[E^x] - ArcTanh[E^x]

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Maple [C]  time = 0.063, size = 36, normalized size = 2.3 \begin{align*}{{\rm e}^{x}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{2}}+{\frac{i}{2}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{2}}\ln \left ({{\rm e}^{x}}+i \right ) -{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)+1/2*ln(exp(x)-1)+1/2*I*ln(exp(x)-I)-1/2*I*ln(exp(x)+I)-1/2*ln(exp(x)+1)

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Maxima [A]  time = 1.53196, size = 30, normalized size = 1.88 \begin{align*} -\arctan \left (e^{x}\right ) + e^{x} - \frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left (e^{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.38938, size = 154, normalized size = 9.62 \begin{align*} -\arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \cosh \left (x\right ) - \frac{1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \coth{\left (2 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x)*coth(2*x), x)

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Giac [A]  time = 1.28272, size = 31, normalized size = 1.94 \begin{align*} -\arctan \left (e^{x}\right ) + e^{x} - \frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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