3.211 \(\int \text {sech}(2 x) \sinh (x) \, dx\)

Optimal. Leaf size=16 \[ -\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}} \]

[Out]

-1/2*arctanh(cosh(x)*2^(1/2))*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4357, 207} \[ -\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[Sech[2*x]*Sinh[x],x]

[Out]

-(ArcTanh[Sqrt[2]*Cosh[x]]/Sqrt[2])

Rule 207

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

Rule 4357

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rubi steps

\begin {align*} \int \text {sech}(2 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{\sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.30, size = 155, normalized size = 9.69 \[ \frac {-4 \tanh ^{-1}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )+\log \left (\sqrt {2}-2 \cosh (x)\right )-\log \left (2 \cosh (x)+\sqrt {2}\right )-2 i \tan ^{-1}\left (\frac {\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (\sqrt {2}-1\right ) \sinh \left (\frac {x}{2}\right )}\right )+2 i \tan ^{-1}\left (\frac {\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )}{\left (\sqrt {2}-1\right ) \cosh \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )}{4 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[2*x]*Sinh[x],x]

[Out]

((-2*I)*ArcTan[(Cosh[x/2] + Sinh[x/2])/((1 + Sqrt[2])*Cosh[x/2] - (-1 + Sqrt[2])*Sinh[x/2])] + (2*I)*ArcTan[(C
osh[x/2] + Sinh[x/2])/((-1 + Sqrt[2])*Cosh[x/2] - (1 + Sqrt[2])*Sinh[x/2])] - 4*ArcTanh[Sqrt[2] - I*Tanh[x/2]]
+ Log[Sqrt[2] - 2*Cosh[x]] - Log[Sqrt[2] + 2*Cosh[x]])/(4*Sqrt[2])

________________________________________________________________________________________

fricas [B]  time = 0.49, size = 35, normalized size = 2.19 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 2 \, \sqrt {2} \cosh \relax (x) + 2}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(2*x)*sinh(x),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log((cosh(x)^2 + sinh(x)^2 - 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + sinh(x)^2))

________________________________________________________________________________________

giac [B]  time = 0.13, size = 38, normalized size = 2.38 \[ -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(2*x)*sinh(x),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.23, size = 39, normalized size = 2.44 \[ \frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{4}-\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(2*x)*sinh(x),x)

[Out]

1/4*ln(1+exp(2*x)-exp(x)*2^(1/2))*2^(1/2)-1/4*ln(1+exp(2*x)+exp(x)*2^(1/2))*2^(1/2)

________________________________________________________________________________________

maxima [B]  time = 0.40, size = 42, normalized size = 2.62 \[ -\frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(2*x)*sinh(x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 35, normalized size = 2.19 \[ -\frac {\sqrt {2}\,\left (\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )-\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2^(1/2)*(log(exp(2*x) + 2^(1/2)*exp(x) + 1) - log(exp(2*x) - 2^(1/2)*exp(x) + 1)))/4

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (2 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(2*x)*sinh(x),x)

[Out]

Integral(sinh(x)*sech(2*x), x)

________________________________________________________________________________________