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]

1/2*arctanh(2*cosh(x)/(2-2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/2*arctanh(2*cosh(x)/(2+2^(1/2))^(1/2))/(4+2*2^(
1/2))^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 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 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 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}(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*}

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Mathematica [C]  time = 0.03, 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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fricas [B]  time = 0.45, size = 215, normalized size = 3.03 \[ \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + {\left ({\left (\sqrt {2} - 1\right )} \cosh \relax (x) + {\left (\sqrt {2} - 1\right )} \sinh \relax (x)\right )} \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - {\left ({\left (\sqrt {2} - 1\right )} \cosh \relax (x) + {\left (\sqrt {2} - 1\right )} \sinh \relax (x)\right )} \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + {\left ({\left (\sqrt {2} + 1\right )} \cosh \relax (x) + {\left (\sqrt {2} + 1\right )} \sinh \relax (x)\right )} \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - {\left ({\left (\sqrt {2} + 1\right )} \cosh \relax (x) + {\left (\sqrt {2} + 1\right )} \sinh \relax (x)\right )} \sqrt {-\sqrt {2} + 2} + 1\right ) \]

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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giac [B]  time = 0.19, size = 115, normalized size = 1.62 \[ -\frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \]

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)

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maple [C]  time = 0.25, size = 40, normalized size = 0.56 \[ 2 \left (\munderset {\textit {\_R} =\RootOf \left (32768 \textit {\_Z}^{4}-512 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (4096 \textit {\_R}^{3}-48 \textit {\_R} \right ) {\mathrm e}^{x}+1\right )\right ) \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}\left (4 \, x\right ) \sinh \relax (x)\,{d x} \]

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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mupad [B]  time = 1.48, size = 251, normalized size = 3.54 \[ \ln \left (3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}+8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+3\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left (3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}-8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+3\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left (3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}-8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+3\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+\ln \left (3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}+8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+3\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/cosh(4*x),x)

[Out]

log(3*exp(2*x) - 2*2^(1/2) + 8*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) - 2*2^(1/2)*exp(2*x) - 8*2^(1/2)*exp(x)*(1/32
- 2^(1/2)/64)^(1/2) + 3)*(1/32 - 2^(1/2)/64)^(1/2) - log(3*exp(2*x) - 2*2^(1/2) - 8*exp(x)*(1/32 - 2^(1/2)/64)
^(1/2) - 2*2^(1/2)*exp(2*x) + 8*2^(1/2)*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) + 3)*(1/32 - 2^(1/2)/64)^(1/2) - log(
3*exp(2*x) + 2*2^(1/2) - 8*exp(x)*(2^(1/2)/64 + 1/32)^(1/2) + 2*2^(1/2)*exp(2*x) - 8*2^(1/2)*exp(x)*(2^(1/2)/6
4 + 1/32)^(1/2) + 3)*(2^(1/2)/64 + 1/32)^(1/2) + log(3*exp(2*x) + 2*2^(1/2) + 8*exp(x)*(2^(1/2)/64 + 1/32)^(1/
2) + 2*2^(1/2)*exp(2*x) + 8*2^(1/2)*exp(x)*(2^(1/2)/64 + 1/32)^(1/2) + 3)*(2^(1/2)/64 + 1/32)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (4 x \right )}\, dx \]

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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