3.10.39 \(\int e^x \text {sech}(2 x) \tanh ^2(2 x) \, dx\) [939]

Optimal. Leaf size=130 \[ \frac {e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac {3 e^{3 x}}{4 \left (1+e^{4 x}\right )}-\frac {5 \text {ArcTan}\left (1-\sqrt {2} e^x\right )}{8 \sqrt {2}}+\frac {5 \text {ArcTan}\left (1+\sqrt {2} e^x\right )}{8 \sqrt {2}}+\frac {5 \log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{16 \sqrt {2}}-\frac {5 \log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{16 \sqrt {2}} \]

[Out]

exp(3*x)/(1+exp(4*x))^2-3/4*exp(3*x)/(1+exp(4*x))+5/16*arctan(-1+exp(x)*2^(1/2))*2^(1/2)+5/16*arctan(1+exp(x)*
2^(1/2))*2^(1/2)+5/32*ln(1+exp(2*x)-exp(x)*2^(1/2))*2^(1/2)-5/32*ln(1+exp(2*x)+exp(x)*2^(1/2))*2^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {2320, 12, 474, 468, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {5 \text {ArcTan}\left (1-\sqrt {2} e^x\right )}{8 \sqrt {2}}+\frac {5 \text {ArcTan}\left (\sqrt {2} e^x+1\right )}{8 \sqrt {2}}-\frac {3 e^{3 x}}{4 \left (e^{4 x}+1\right )}+\frac {e^{3 x}}{\left (e^{4 x}+1\right )^2}+\frac {5 \log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{16 \sqrt {2}}-\frac {5 \log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{16 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 210

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

Rule 303

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

Rule 468

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 474

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 631

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 642

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 1176

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 1179

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.05, size = 58, normalized size = 0.45 \begin {gather*} \frac {e^{3 x}-3 e^{7 x}}{4 \left (1+e^{4 x}\right )^2}-\frac {5}{16} \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {x-\log \left (e^x-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(3*x) - 3*E^(7*x))/(4*(1 + E^(4*x))^2) - (5*RootSum[1 + #1^4 & , (x - Log[E^x - #1])/#1 & ])/16

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.56, size = 48, normalized size = 0.37

method	result	size
risch	\(-\frac {{\mathrm e}^{3 x} \left (3 \,{\mathrm e}^{4 x}-1\right )}{4 \left (1+{\mathrm e}^{4 x}\right )^{2}}+2 \left (\munderset {\textit {\_R} =\RootOf \left (1048576 \textit {\_Z}^{4}+625\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+\frac {32768 \textit {\_R}^{3}}{125}\right )\right )\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*sech(2*x)*tanh(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4*exp(3*x)*(3*exp(4*x)-1)/(1+exp(4*x))^2+2*sum(_R*ln(exp(x)+32768/125*_R^3),_R=RootOf(1048576*_Z^4+625))

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Maxima [A]
time = 0.48, size = 105, normalized size = 0.81 \begin {gather*} \frac {5}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) + \frac {5}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {5}{32} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {5}{32} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {3 \, e^{\left (7 \, x\right )} - e^{\left (3 \, x\right )}}{4 \, {\left (e^{\left (8 \, x\right )} + 2 \, e^{\left (4 \, x\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) + 5/16*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 5/3
2*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 5/32*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/4*(3*e^(7*x) - e^(
3*x))/(e^(8*x) + 2*e^(4*x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (93) = 186\).
time = 0.35, size = 213, normalized size = 1.64 \begin {gather*} -\frac {20 \, {\left (\sqrt {2} e^{\left (8 \, x\right )} + 2 \, \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 ) + 20 \, {\left (\sqrt {2} e^{\left (8 \, x\right )} + 2 \, \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 ) + 5 \, {\left (\sqrt {2} e^{\left (8 \, x\right )} + 2 \, \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 ) - 5 \, {\left (\sqrt {2} e^{\left (8 \, x\right )} + 2 \, \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 ) + 24 \, e^{\left (7 \, x\right )} - 8 \, e^{\left (3 \, x\right )}}{32 \, {\left (e^{\left (8 \, x\right )} + 2 \, e^{\left (4 \, x\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(20*(sqrt(2)*e^(8*x) + 2*sqrt(2)*e^(4*x) + sqrt(2))*arctan(-sqrt(2)*e^x + sqrt(2)*sqrt(sqrt(2)*e^x + e^(
2*x) + 1) - 1) + 20*(sqrt(2)*e^(8*x) + 2*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) + 5*(sqrt(2)*e^(8*x) + 2*sqrt(2)*e^(4*x) + sqrt(2))*log(4*sqrt(2)*e^x + 4*e^
(2*x) + 4) - 5*(sqrt(2)*e^(8*x) + 2*sqrt(2)*e^(4*x) + sqrt(2))*log(-4*sqrt(2)*e^x + 4*e^(2*x) + 4) + 24*e^(7*x
) - 8*e^(3*x))/(e^(8*x) + 2*e^(4*x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \tanh ^{2}{\left (2 x \right )} \operatorname {sech}{\left (2 x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 99, normalized size = 0.76 \begin {gather*} \frac {5}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) + \frac {5}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {5}{32} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {5}{32} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {3 \, e^{\left (7 \, x\right )} - e^{\left (3 \, x\right )}}{4 \, {\left (e^{\left (4 \, x\right )} + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) + 5/16*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 5/3
2*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) + 5/32*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) - 1/4*(3*e^(7*x) - e^(
3*x))/(e^(4*x) + 1)^2

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Mupad [B]
time = 2.09, size = 112, normalized size = 0.86 \begin {gather*} \frac {{\mathrm {e}}^{3\,x}}{2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {3\,{\mathrm {e}}^{3\,x}}{4\,\left ({\mathrm {e}}^{4\,x}+1\right )}+\sqrt {2}\,\ln \left (\frac {25}{16}+\sqrt {2}\,{\mathrm {e}}^x\,\left (-\frac {25}{32}-\frac {25}{32}{}\mathrm {i}\right )\right )\,\left (\frac {5}{32}+\frac {5}{32}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (\frac {25}{16}+\sqrt {2}\,{\mathrm {e}}^x\,\left (-\frac {25}{32}+\frac {25}{32}{}\mathrm {i}\right )\right )\,\left (\frac {5}{32}-\frac {5}{32}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (\frac {25}{16}+\sqrt {2}\,{\mathrm {e}}^x\,\left (\frac {25}{32}-\frac {25}{32}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{32}+\frac {5}{32}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (\frac {25}{16}+\sqrt {2}\,{\mathrm {e}}^x\,\left (\frac {25}{32}+\frac {25}{32}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{32}-\frac {5}{32}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((tanh(2*x)^2*exp(x))/cosh(2*x),x)

[Out]

2^(1/2)*log(25/16 - 2^(1/2)*exp(x)*(25/32 + 25i/32))*(5/32 + 5i/32) + 2^(1/2)*log(25/16 - 2^(1/2)*exp(x)*(25/3
2 - 25i/32))*(5/32 - 5i/32) - 2^(1/2)*log(2^(1/2)*exp(x)*(25/32 - 25i/32) + 25/16)*(5/32 - 5i/32) - 2^(1/2)*lo
g(2^(1/2)*exp(x)*(25/32 + 25i/32) + 25/16)*(5/32 + 5i/32) + exp(3*x)/(2*exp(4*x) + exp(8*x) + 1) - (3*exp(3*x)
)/(4*(exp(4*x) + 1))

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