3.4.12 \(\int e^x \text {csch}(2 x) \, dx\) [312]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 11, 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 = {2320, 12, 304, 209, 212} \begin {gather*} \text {ArcTan}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[E^x] - ArcTanh[E^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 209

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

Rule 212

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

Rule 304

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 2320

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

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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \end {gather*}

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] Result contains complex when optimal does not.
time = 0.58, size = 34, normalized size = 3.09

method	result	size
risch	\(\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*csch(2*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.48, size = 18, normalized size = 1.64 \begin {gather*} \arctan \left (e^{x}\right ) - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (9) = 18\).
time = 0.39, size = 25, normalized size = 2.27 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \operatorname {csch}{\left (2 x \right )}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (9) = 18\).
time = 0.40, size = 19, normalized size = 1.73 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 0.17, size = 26, normalized size = 2.36 \begin {gather*} \frac {\ln \left (4\,{\mathrm {e}}^x-4\right )}{2}-\frac {\ln \left (-4\,{\mathrm {e}}^x-4\right )}{2}-\mathrm {atan}\left ({\mathrm {e}}^{-x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/sinh(2*x),x)

[Out]

log(4*exp(x) - 4)/2 - log(- 4*exp(x) - 4)/2 - atan(exp(-x))

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