3.312 \(\int e^x \text{csch}(2 x) \, dx\)

Optimal. Leaf size=11 \[ \tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

[Out]

ArcTan[E^x] - ArcTanh[E^x]

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Rubi [A]  time = 0.0127131, antiderivative size = 11, 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, 12, 298, 203, 206} \[ \tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 298

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

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

Mathematica [A]  time = 0.0091644, size = 11, normalized size = 1. \[ \tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[E^x] - ArcTanh[E^x]

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Maple [C]  time = 0.045, size = 34, normalized size = 3.1 \begin{align*}{\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)*csch(2*x),x)

[Out]

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 = 1.5826, size = 24, normalized size = 2.18 \begin{align*} \arctan \left (e^{x}\right ) - \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)*csch(2*x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.06641, size = 126, normalized size = 11.45 \begin{align*} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(x)*csch(2*x), x)

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Giac [B]  time = 1.1461, size = 26, normalized size = 2.36 \begin{align*} \arctan \left (e^{x}\right ) - \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)*csch(2*x),x, algorithm="giac")

[Out]

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