### 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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0194854, antiderivative size = 16, normalized size of antiderivative = 1., 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 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])

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])

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 = 0.28133, 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])

________________________________________________________________________________________

Maple [B]  time = 0.033, size = 39, normalized size = 2.4 \begin{align*}{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}-{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{4}}-{\frac{\ln \left ( 1+{{\rm e}^{2\,x}}+{{\rm e}^{x}}\sqrt{2} \right ) \sqrt{2}}{4}} \end{align*}

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 = 1.52406, size = 57, normalized size = 3.56 \begin{align*} -\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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 1.98553, size = 122, normalized size = 7.62 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) \end{align*}

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [B]  time = 1.19838, size = 51, normalized size = 3.19 \begin{align*} -\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 ) \end{align*}

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)