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]

exp(x)-arctan(exp(x))-arctanh(exp(x))

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 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])

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 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 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 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]]]

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ e^x-\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

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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fricas [B]  time = 0.51, size = 31, normalized size = 1.94 \[ -\arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + \cosh \relax (x) - \frac {1}{2} \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + \sinh \relax (x) \]

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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giac [A]  time = 0.13, size = 23, normalized size = 1.44 \[ -\arctan \left (e^{x}\right ) + e^{x} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

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))

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maple [C]  time = 0.20, size = 36, normalized size = 2.25 \[ {\mathrm e}^{x}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 22, normalized size = 1.38 \[ -\arctan \left (e^{x}\right ) + e^{x} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]

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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mupad [B]  time = 0.17, size = 26, normalized size = 1.62 \[ \frac {\ln \left (2-2\,{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-2\,{\mathrm {e}}^x-2\right )}{2}-\mathrm {atan}\left ({\mathrm {e}}^x\right )+{\mathrm {e}}^x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2 - 2*exp(x))/2 - log(- 2*exp(x) - 2)/2 - atan(exp(x)) + exp(x)

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

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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