3.224 \(\int e^x \coth (4 x) \, dx\)

Optimal. Leaf size=116 \[ e^x+\frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{4 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{4 \sqrt{2}}-\frac{1}{2} \tan ^{-1}\left (e^x\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left (e^x\right ) \]

[Out]

E^x - ArcTan[E^x]/2 + ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTanh[E^x]
/2 + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])

________________________________________________________________________________________

Rubi [A]  time = 0.0777028, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.5, Rules used = {2282, 388, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ e^x+\frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{4 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{4 \sqrt{2}}-\frac{1}{2} \tan ^{-1}\left (e^x\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Coth[4*x],x]

[Out]

E^x - ArcTan[E^x]/2 + ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTanh[E^x]
/2 + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])

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 214

Int[((a_) + (b_.)*(x_)^(n_))^(-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^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a,
b}, x] && IGtQ[n/4, 1] &&  !GtQ[a/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 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])

Rule 211

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int e^x \coth (4 x) \, dx &=\operatorname{Subst}\left (\int \frac{-1-x^8}{1-x^8} \, dx,x,e^x\right )\\ &=e^x-2 \operatorname{Subst}\left (\int \frac{1}{1-x^8} \, dx,x,e^x\right )\\ &=e^x-\operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-\frac{1}{2} \tan ^{-1}\left (e^x\right )-\frac{1}{2} \tanh ^{-1}\left (e^x\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,e^x\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,e^x\right )}{4 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,e^x\right )}{4 \sqrt{2}}\\ &=e^x-\frac{1}{2} \tan ^{-1}\left (e^x\right )-\frac{1}{2} \tanh ^{-1}\left (e^x\right )+\frac{\log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{4 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{4 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} e^x\right )}{2 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} e^x\right )}{2 \sqrt{2}}\\ &=e^x-\frac{1}{2} \tan ^{-1}\left (e^x\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} e^x\right )}{2 \sqrt{2}}-\frac{1}{2} \tanh ^{-1}\left (e^x\right )+\frac{\log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{4 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0164104, size = 22, normalized size = 0.19 \[ e^x-2 e^x \, _2F_1\left (\frac{1}{8},1;\frac{9}{8};e^{8 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Coth[4*x],x]

[Out]

E^x - 2*E^x*Hypergeometric2F1[1/8, 1, 9/8, E^(8*x)]

________________________________________________________________________________________

Maple [C]  time = 0.07, size = 56, normalized size = 0.5 \begin{align*}{{\rm e}^{x}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{4}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{4}}+{\frac{i}{4}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{4}}\ln \left ({{\rm e}^{x}}+i \right ) +\sum _{{\it \_R}={\it RootOf} \left ( 256\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{x}}-4\,{\it \_R} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*coth(4*x),x)

[Out]

exp(x)-1/4*ln(exp(x)+1)+1/4*ln(exp(x)-1)+1/4*I*ln(exp(x)-I)-1/4*I*ln(exp(x)+I)+sum(_R*ln(exp(x)-4*_R),_R=RootO
f(256*_Z^4+1))

________________________________________________________________________________________

Maxima [A]  time = 1.69206, size = 131, normalized size = 1.13 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{8} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac{1}{4} \, \log \left (e^{x} + 1\right ) + \frac{1}{4} \, \log \left (e^{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(4*x),x, algorithm="maxima")

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) - 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 1/8*
sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 1/8*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/2*arctan(e^x) + e^x -
1/4*log(e^x + 1) + 1/4*log(e^x - 1)

________________________________________________________________________________________

Fricas [A]  time = 2.10871, size = 447, normalized size = 3.85 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (-\sqrt{2} e^{x} + \sqrt{2} \sqrt{\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\sqrt{2} e^{x} + \frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + 1\right ) - \frac{1}{8} \, \sqrt{2} \log \left (4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac{1}{8} \, \sqrt{2} \log \left (-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) - \frac{1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac{1}{4} \, \log \left (e^{x} + 1\right ) + \frac{1}{4} \, \log \left (e^{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(4*x),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(-sqrt(2)*e^x + sqrt(2)*sqrt(sqrt(2)*e^x + e^(2*x) + 1) - 1) + 1/2*sqrt(2)*arctan(-sqrt(2)*e
^x + 1/2*sqrt(2)*sqrt(-4*sqrt(2)*e^x + 4*e^(2*x) + 4) + 1) - 1/8*sqrt(2)*log(4*sqrt(2)*e^x + 4*e^(2*x) + 4) +
1/8*sqrt(2)*log(-4*sqrt(2)*e^x + 4*e^(2*x) + 4) - 1/2*arctan(e^x) + e^x - 1/4*log(e^x + 1) + 1/4*log(e^x - 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \coth{\left (4 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*coth(4*x),x)

[Out]

Integral(exp(x)*coth(4*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.3042, size = 132, normalized size = 1.14 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{8} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{8} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac{1}{4} \, \log \left (e^{x} + 1\right ) + \frac{1}{4} \, \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(4*x),x, algorithm="giac")

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) - 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 1/8*
sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 1/8*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/2*arctan(e^x) + e^x -
1/4*log(e^x + 1) + 1/4*log(abs(e^x - 1))