3.219 \(\int e^x \tanh (3 x) \, dx\)

Optimal. Leaf size=97 \[ e^x+\frac{\log \left (-\sqrt{3} e^x+e^{2 x}+1\right )}{2 \sqrt{3}}-\frac{\log \left (\sqrt{3} e^x+e^{2 x}+1\right )}{2 \sqrt{3}}-\frac{2}{3} \tan ^{-1}\left (e^x\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 e^x\right )-\frac{1}{3} \tan ^{-1}\left (2 e^x+\sqrt{3}\right ) \]

[Out]

E^x - (2*ArcTan[E^x])/3 + ArcTan[Sqrt[3] - 2*E^x]/3 - ArcTan[Sqrt[3] + 2*E^x]/3 + Log[1 - Sqrt[3]*E^x + E^(2*x
)]/(2*Sqrt[3]) - Log[1 + Sqrt[3]*E^x + E^(2*x)]/(2*Sqrt[3])

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Rubi [A]  time = 0.186301, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2282, 388, 209, 634, 618, 204, 628, 203} \[ e^x+\frac{\log \left (-\sqrt{3} e^x+e^{2 x}+1\right )}{2 \sqrt{3}}-\frac{\log \left (\sqrt{3} e^x+e^{2 x}+1\right )}{2 \sqrt{3}}-\frac{2}{3} \tan ^{-1}\left (e^x\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 e^x\right )-\frac{1}{3} \tan ^{-1}\left (2 e^x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*Tanh[3*x],x]

[Out]

E^x - (2*ArcTan[E^x])/3 + ArcTan[Sqrt[3] - 2*E^x]/3 - ArcTan[Sqrt[3] + 2*E^x]/3 + Log[1 - Sqrt[3]*E^x + E^(2*x
)]/(2*Sqrt[3]) - Log[1 + Sqrt[3]*E^x + E^(2*x)]/(2*Sqrt[3])

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 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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])

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 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 \tanh (3 x) \, dx &=\operatorname{Subst}\left (\int \frac{-1+x^6}{1+x^6} \, dx,x,e^x\right )\\ &=e^x-2 \operatorname{Subst}\left (\int \frac{1}{1+x^6} \, dx,x,e^x\right )\\ &=e^x-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^x\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,e^x\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,e^x\right )\\ &=e^x-\frac{2}{3} \tan ^{-1}\left (e^x\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,e^x\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,e^x\right )+\frac{\operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,e^x\right )}{2 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,e^x\right )}{2 \sqrt{3}}\\ &=e^x-\frac{2}{3} \tan ^{-1}\left (e^x\right )+\frac{\log \left (1-\sqrt{3} e^x+e^{2 x}\right )}{2 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} e^x+e^{2 x}\right )}{2 \sqrt{3}}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 e^x\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 e^x\right )\\ &=e^x-\frac{2}{3} \tan ^{-1}\left (e^x\right )+\frac{1}{3} \tan ^{-1}\left (\sqrt{3}-2 e^x\right )-\frac{1}{3} \tan ^{-1}\left (\sqrt{3}+2 e^x\right )+\frac{\log \left (1-\sqrt{3} e^x+e^{2 x}\right )}{2 \sqrt{3}}-\frac{\log \left (1+\sqrt{3} e^x+e^{2 x}\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0117698, size = 24, normalized size = 0.25 \[ e^x-2 e^x \, _2F_1\left (\frac{1}{6},1;\frac{7}{6};-e^{6 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Tanh[3*x],x]

[Out]

E^x - 2*E^x*Hypergeometric2F1[1/6, 1, 7/6, -E^(6*x)]

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Maple [C]  time = 0.071, size = 47, normalized size = 0.5 \begin{align*}{{\rm e}^{x}}+{\frac{i}{3}}\ln \left ({{\rm e}^{x}}-i \right ) -{\frac{i}{3}}\ln \left ({{\rm e}^{x}}+i \right ) +\sum _{{\it \_R}={\it RootOf} \left ( 81\,{{\it \_Z}}^{4}-9\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{x}}-3\,{\it \_R} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*tanh(3*x),x)

[Out]

exp(x)+1/3*I*ln(exp(x)-I)-1/3*I*ln(exp(x)+I)+sum(_R*ln(exp(x)-3*_R),_R=RootOf(81*_Z^4-9*_Z^2+1))

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Maxima [A]  time = 1.58983, size = 93, normalized size = 0.96 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{6} \, \sqrt{3} \log \left (-\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{3} \, \arctan \left (\sqrt{3} + 2 \, e^{x}\right ) - \frac{1}{3} \, \arctan \left (-\sqrt{3} + 2 \, e^{x}\right ) - \frac{2}{3} \, \arctan \left (e^{x}\right ) + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tanh(3*x),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*log(sqrt(3)*e^x + e^(2*x) + 1) + 1/6*sqrt(3)*log(-sqrt(3)*e^x + e^(2*x) + 1) - 1/3*arctan(sqrt(3)
+ 2*e^x) - 1/3*arctan(-sqrt(3) + 2*e^x) - 2/3*arctan(e^x) + e^x

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Fricas [A]  time = 2.29901, size = 350, normalized size = 3.61 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac{1}{6} \, \sqrt{3} \log \left (-4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac{2}{3} \, \arctan \left (\sqrt{3} + \sqrt{-4 \, \sqrt{3} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} - 2 \, e^{x}\right ) + \frac{2}{3} \, \arctan \left (-\sqrt{3} + 2 \, \sqrt{\sqrt{3} e^{x} + e^{\left (2 \, x\right )} + 1} - 2 \, e^{x}\right ) - \frac{2}{3} \, \arctan \left (e^{x}\right ) + e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tanh(3*x),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*log(4*sqrt(3)*e^x + 4*e^(2*x) + 4) + 1/6*sqrt(3)*log(-4*sqrt(3)*e^x + 4*e^(2*x) + 4) + 2/3*arctan
(sqrt(3) + sqrt(-4*sqrt(3)*e^x + 4*e^(2*x) + 4) - 2*e^x) + 2/3*arctan(-sqrt(3) + 2*sqrt(sqrt(3)*e^x + e^(2*x)
+ 1) - 2*e^x) - 2/3*arctan(e^x) + e^x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \tanh{\left (3 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tanh(3*x),x)

[Out]

Integral(exp(x)*tanh(3*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \tanh \left (3 \, x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*tanh(3*x),x, algorithm="giac")

[Out]

integrate(e^x*tanh(3*x), x)