3.213 \(\int \text{sech}(4 x) \sinh (x) \, dx\)

Optimal. Leaf size=71 \[ \frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

[Out]

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

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Rubi [A]  time = 0.0735708, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4357, 1093, 207} \[ \frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[Sech[4*x]*Sinh[x],x]

[Out]

ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[2]]]/(2*S
qrt[2*(2 + 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 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

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}(4 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-8 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-4-2 \sqrt{2}+8 x^2} \, dx,x,\cosh (x)\right )-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-4+2 \sqrt{2}+8 x^2} \, dx,x,\cosh (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{2}}}\right )}{2 \sqrt{2 \left (2-\sqrt{2}\right )}}-\frac{\tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2 \left (2+\sqrt{2}\right )}}\\ \end{align*}

Mathematica [C]  time = 0.0237442, size = 110, normalized size = 1.55 \[ \frac{1}{16} \text{RootSum}\left [\text{$\#$1}^8+1\& ,\frac{\text{$\#$1}^2 x+2 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-x}{\text{$\#$1}^5}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[4*x]*Sinh[x],x]

[Out]

RootSum[1 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + x*#1^2 + 2*Log[-Cosh[
x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2)/#1^5 & ]/16

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Maple [C]  time = 0.046, size = 40, normalized size = 0.6 \begin{align*} 2\,\sum _{{\it \_R}={\it RootOf} \left ( 32768\,{{\it \_Z}}^{4}-512\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{2\,x}}+ \left ( 4096\,{{\it \_R}}^{3}-48\,{\it \_R} \right ){{\rm e}^{x}}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(4*x)*sinh(x),x)

[Out]

2*sum(_R*ln(exp(2*x)+(4096*_R^3-48*_R)*exp(x)+1),_R=RootOf(32768*_Z^4-512*_Z^2+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (4 \, x\right ) \sinh \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(4*x)*sinh(x),x, algorithm="maxima")

[Out]

integrate(sech(4*x)*sinh(x), x)

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Fricas [B]  time = 2.16482, size = 776, normalized size = 10.93 \begin{align*} \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} +{\left ({\left (\sqrt{2} - 1\right )} \cosh \left (x\right ) +{\left (\sqrt{2} - 1\right )} \sinh \left (x\right )\right )} \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} -{\left ({\left (\sqrt{2} - 1\right )} \cosh \left (x\right ) +{\left (\sqrt{2} - 1\right )} \sinh \left (x\right )\right )} \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} +{\left ({\left (\sqrt{2} + 1\right )} \cosh \left (x\right ) +{\left (\sqrt{2} + 1\right )} \sinh \left (x\right )\right )} \sqrt{-\sqrt{2} + 2} + 1\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} -{\left ({\left (\sqrt{2} + 1\right )} \cosh \left (x\right ) +{\left (\sqrt{2} + 1\right )} \sinh \left (x\right )\right )} \sqrt{-\sqrt{2} + 2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(4*x)*sinh(x),x, algorithm="fricas")

[Out]

1/8*sqrt(sqrt(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(2) - 1)*cosh(x) + (sqrt(2) - 1)*s
inh(x))*sqrt(sqrt(2) + 2) + 1) - 1/8*sqrt(sqrt(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(
2) - 1)*cosh(x) + (sqrt(2) - 1)*sinh(x))*sqrt(sqrt(2) + 2) + 1) - 1/8*sqrt(-sqrt(2) + 2)*log(cosh(x)^2 + 2*cos
h(x)*sinh(x) + sinh(x)^2 + ((sqrt(2) + 1)*cosh(x) + (sqrt(2) + 1)*sinh(x))*sqrt(-sqrt(2) + 2) + 1) + 1/8*sqrt(
-sqrt(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(2) + 1)*cosh(x) + (sqrt(2) + 1)*sinh(x))*
sqrt(-sqrt(2) + 2) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \operatorname{sech}{\left (4 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(4*x)*sinh(x),x)

[Out]

Integral(sinh(x)*sech(4*x), x)

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Giac [B]  time = 1.22065, size = 155, normalized size = 2.18 \begin{align*} -\frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left (\sqrt{\sqrt{2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left (-\sqrt{\sqrt{2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (\sqrt{-\sqrt{2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (-\sqrt{-\sqrt{2} + 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(4*x)*sinh(x),x, algorithm="giac")

[Out]

-1/8*sqrt(-sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/8*sqrt(-sqrt(2) + 2)*log(-sqrt(sqrt(2) +
2)*e^x + e^(2*x) + 1) + 1/8*sqrt(sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) - 1/8*sqrt(sqrt(2) + 2
)*log(-sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1)