3.940 \(\int e^x \text{sech}^2(2 x) \tanh ^2(2 x) \, dx\)

Optimal. Leaf size=149 \[ -\frac{3 e^x}{8 \left (e^{4 x}+1\right )}-\frac{5 e^{5 x}}{6 \left (e^{4 x}+1\right )^2}+\frac{4 e^{5 x}}{3 \left (e^{4 x}+1\right )^3}-\frac{3 \log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{32 \sqrt{2}}+\frac{3 \log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{32 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} e^x\right )}{16 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} e^x+1\right )}{16 \sqrt{2}} \]

[Out]

(4*E^(5*x))/(3*(1 + E^(4*x))^3) - (5*E^(5*x))/(6*(1 + E^(4*x))^2) - (3*E^x)/(8*(1 + E^(4*x))) - (3*ArcTan[1 -
Sqrt[2]*E^x])/(16*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*E^x])/(16*Sqrt[2]) - (3*Log[1 - Sqrt[2]*E^x + E^(2*x)])/(32
*Sqrt[2]) + (3*Log[1 + Sqrt[2]*E^x + E^(2*x)])/(32*Sqrt[2])

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Rubi [A]  time = 0.127663, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {2282, 12, 463, 457, 288, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 e^x}{8 \left (e^{4 x}+1\right )}-\frac{5 e^{5 x}}{6 \left (e^{4 x}+1\right )^2}+\frac{4 e^{5 x}}{3 \left (e^{4 x}+1\right )^3}-\frac{3 \log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{32 \sqrt{2}}+\frac{3 \log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{32 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} e^x\right )}{16 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} e^x+1\right )}{16 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[E^x*Sech[2*x]^2*Tanh[2*x]^2,x]

[Out]

(4*E^(5*x))/(3*(1 + E^(4*x))^3) - (5*E^(5*x))/(6*(1 + E^(4*x))^2) - (3*E^x)/(8*(1 + E^(4*x))) - (3*ArcTan[1 -
Sqrt[2]*E^x])/(16*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*E^x])/(16*Sqrt[2]) - (3*Log[1 - Sqrt[2]*E^x + E^(2*x)])/(32
*Sqrt[2]) + (3*Log[1 + Sqrt[2]*E^x + E^(2*x)])/(32*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 12

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

Rule 463

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> -Simp[((b*c - a*
d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b^2*e*n*(p + 1)), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a + b
*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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 \text{sech}^2(2 x) \tanh ^2(2 x) \, dx &=\operatorname{Subst}\left (\int \frac{4 x^4 \left (1-x^4\right )^2}{\left (1+x^4\right )^4} \, dx,x,e^x\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^4 \left (1-x^4\right )^2}{\left (1+x^4\right )^4} \, dx,x,e^x\right )\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4 \left (8-12 x^4\right )}{\left (1+x^4\right )^3} \, dx,x,e^x\right )\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{5 e^{5 x}}{6 \left (1+e^{4 x}\right )^2}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^4\right )^2} \, dx,x,e^x\right )\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{5 e^{5 x}}{6 \left (1+e^{4 x}\right )^2}-\frac{3 e^x}{8 \left (1+e^{4 x}\right )}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,e^x\right )\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{5 e^{5 x}}{6 \left (1+e^{4 x}\right )^2}-\frac{3 e^x}{8 \left (1+e^{4 x}\right )}+\frac{3}{16} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,e^x\right )+\frac{3}{16} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{5 e^{5 x}}{6 \left (1+e^{4 x}\right )^2}-\frac{3 e^x}{8 \left (1+e^{4 x}\right )}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,e^x\right )-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,e^x\right )}{32 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,e^x\right )}{32 \sqrt{2}}\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{5 e^{5 x}}{6 \left (1+e^{4 x}\right )^2}-\frac{3 e^x}{8 \left (1+e^{4 x}\right )}-\frac{3 \log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{32 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{32 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} e^x\right )}{16 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} e^x\right )}{16 \sqrt{2}}\\ &=\frac{4 e^{5 x}}{3 \left (1+e^{4 x}\right )^3}-\frac{5 e^{5 x}}{6 \left (1+e^{4 x}\right )^2}-\frac{3 e^x}{8 \left (1+e^{4 x}\right )}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} e^x\right )}{16 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} e^x\right )}{16 \sqrt{2}}-\frac{3 \log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{32 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{32 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0621056, size = 64, normalized size = 0.43 \[ \frac{1}{96} \left (-9 \text{RootSum}\left [\text{$\#$1}^4+1\& ,\frac{x-\log \left (e^x-\text{$\#$1}\right )}{\text{$\#$1}^3}\& \right ]-\frac{4 e^x \left (6 e^{4 x}+29 e^{8 x}+9\right )}{\left (e^{4 x}+1\right )^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*Sech[2*x]^2*Tanh[2*x]^2,x]

[Out]

((-4*E^x*(9 + 6*E^(4*x) + 29*E^(8*x)))/(1 + E^(4*x))^3 - 9*RootSum[1 + #1^4 & , (x - Log[E^x - #1])/#1^3 & ])/
96

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Maple [C]  time = 0.067, size = 50, normalized size = 0.3 \begin{align*} -{\frac{{{\rm e}^{x}} \left ( 29\,{{\rm e}^{8\,x}}+6\,{{\rm e}^{4\,x}}+9 \right ) }{24\, \left ( 1+{{\rm e}^{4\,x}} \right ) ^{3}}}+4\,\sum _{{\it \_R}={\it RootOf} \left ( 268435456\,{{\it \_Z}}^{4}+81 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{x}}+{\frac{128\,{\it \_R}}{3}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sech(2*x)^2*tanh(2*x)^2,x)

[Out]

-1/24*exp(x)*(29*exp(8*x)+6*exp(4*x)+9)/(1+exp(4*x))^3+4*sum(_R*ln(exp(x)+128/3*_R),_R=RootOf(268435456*_Z^4+8
1))

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Maxima [A]  time = 1.6337, size = 155, normalized size = 1.04 \begin{align*} \frac{3}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{3}{32} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{3}{64} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{29 \, e^{\left (9 \, x\right )} + 6 \, e^{\left (5 \, x\right )} + 9 \, e^{x}}{24 \,{\left (e^{\left (12 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(2*x)^2*tanh(2*x)^2,x, algorithm="maxima")

[Out]

3/32*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) + 3/32*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) + 3/6
4*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) - 3/64*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/24*(29*e^(9*x) + 6
*e^(5*x) + 9*e^x)/(e^(12*x) + 3*e^(8*x) + 3*e^(4*x) + 1)

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Fricas [B]  time = 2.24074, size = 798, normalized size = 5.36 \begin{align*} -\frac{36 \,{\left (\sqrt{2} e^{\left (12 \, x\right )} + 3 \, \sqrt{2} e^{\left (8 \, x\right )} + 3 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} e^{x} + \sqrt{2} \sqrt{\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) + 36 \,{\left (\sqrt{2} e^{\left (12 \, x\right )} + 3 \, \sqrt{2} e^{\left (8 \, x\right )} + 3 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \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 ) - 9 \,{\left (\sqrt{2} e^{\left (12 \, x\right )} + 3 \, \sqrt{2} e^{\left (8 \, x\right )} + 3 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \log \left (4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + 9 \,{\left (\sqrt{2} e^{\left (12 \, x\right )} + 3 \, \sqrt{2} e^{\left (8 \, x\right )} + 3 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \log \left (-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + 232 \, e^{\left (9 \, x\right )} + 48 \, e^{\left (5 \, x\right )} + 72 \, e^{x}}{192 \,{\left (e^{\left (12 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(2*x)^2*tanh(2*x)^2,x, algorithm="fricas")

[Out]

-1/192*(36*(sqrt(2)*e^(12*x) + 3*sqrt(2)*e^(8*x) + 3*sqrt(2)*e^(4*x) + sqrt(2))*arctan(-sqrt(2)*e^x + sqrt(2)*
sqrt(sqrt(2)*e^x + e^(2*x) + 1) - 1) + 36*(sqrt(2)*e^(12*x) + 3*sqrt(2)*e^(8*x) + 3*sqrt(2)*e^(4*x) + sqrt(2))
*arctan(-sqrt(2)*e^x + 1/2*sqrt(2)*sqrt(-4*sqrt(2)*e^x + 4*e^(2*x) + 4) + 1) - 9*(sqrt(2)*e^(12*x) + 3*sqrt(2)
*e^(8*x) + 3*sqrt(2)*e^(4*x) + sqrt(2))*log(4*sqrt(2)*e^x + 4*e^(2*x) + 4) + 9*(sqrt(2)*e^(12*x) + 3*sqrt(2)*e
^(8*x) + 3*sqrt(2)*e^(4*x) + sqrt(2))*log(-4*sqrt(2)*e^x + 4*e^(2*x) + 4) + 232*e^(9*x) + 48*e^(5*x) + 72*e^x)
/(e^(12*x) + 3*e^(8*x) + 3*e^(4*x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(2*x)**2*tanh(2*x)**2,x)

[Out]

Integral(exp(x)*tanh(2*x)**2*sech(2*x)**2, x)

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Giac [A]  time = 1.13499, size = 139, normalized size = 0.93 \begin{align*} \frac{3}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{3}{32} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{3}{64} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{29 \, e^{\left (9 \, x\right )} + 6 \, e^{\left (5 \, x\right )} + 9 \, e^{x}}{24 \,{\left (e^{\left (4 \, x\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*sech(2*x)^2*tanh(2*x)^2,x, algorithm="giac")

[Out]

3/32*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) + 3/32*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) + 3/6
4*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) - 3/64*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/24*(29*e^(9*x) + 6
*e^(5*x) + 9*e^x)/(e^(4*x) + 1)^3