3.215 \(\int \text {sech}(6 x) \sinh (x) \, dx\)

Optimal. Leaf size=85 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]

[Out]

1/6*arctanh(cosh(x)*2^(1/2))*2^(1/2)-1/6*arctanh(2*cosh(x)/(1/2*6^(1/2)-1/2*2^(1/2)))/(1/2*6^(1/2)-1/2*2^(1/2)
)-1/6*arctanh(2*cosh(x)/(1/2*6^(1/2)+1/2*2^(1/2)))/(1/2*6^(1/2)+1/2*2^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4357, 2057, 207, 1166} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]

Antiderivative was successfully verified.

[In]

Int[Sech[6*x]*Sinh[x],x]

[Out]

ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) - ArcTanh[
(2*Cosh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])

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 1166

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

Rule 2057

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(u /. x -> x^2)^p, x], x
] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x^2] && ILtQ[p, 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}(6 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{-1+18 x^2-48 x^4+32 x^6} \, dx,x,\cosh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{3 \left (-1+2 x^2\right )}+\frac {4 \left (-1+2 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\cosh (x)\right )\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {-1+2 x^2}{1-16 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-8-4 \sqrt {3}+16 x^2} \, dx,x,\cosh (x)\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-8+4 \sqrt {3}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.28, size = 385, normalized size = 4.53 \[ \frac {\sqrt {2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^6 x+2 \text {$\#$1}^6 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-\text {$\#$1}^4 x-2 \text {$\#$1}^4 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )+\text {$\#$1}^2 x+2 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-x}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]+8 \tanh ^{-1}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )-2 \log \left (\sqrt {2}-2 \cosh (x)\right )+2 \log \left (2 \cosh (x)+\sqrt {2}\right )+4 i \tan ^{-1}\left (\frac {\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cosh \left (\frac {x}{2}\right )-\left (\sqrt {2}-1\right ) \sinh \left (\frac {x}{2}\right )}\right )-4 i \tan ^{-1}\left (\frac {\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right )}{\left (\sqrt {2}-1\right ) \cosh \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sinh \left (\frac {x}{2}\right )}\right )}{24 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[6*x]*Sinh[x],x]

[Out]

((4*I)*ArcTan[(Cosh[x/2] + Sinh[x/2])/((1 + Sqrt[2])*Cosh[x/2] - (-1 + Sqrt[2])*Sinh[x/2])] - (4*I)*ArcTan[(Co
sh[x/2] + Sinh[x/2])/((-1 + Sqrt[2])*Cosh[x/2] - (1 + Sqrt[2])*Sinh[x/2])] + 8*ArcTanh[Sqrt[2] - I*Tanh[x/2]]
- 2*Log[Sqrt[2] - 2*Cosh[x]] + 2*Log[Sqrt[2] + 2*Cosh[x]] + Sqrt[2]*RootSum[1 - #1^4 + #1^8 & , (-x - 2*Log[-C
osh[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 - S
inh[x/2]*#1]*#1^2 - x*#1^4 - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 + x*#1^6 + 2*Log
[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-#1^3 + 2*#1^7) & ])/(24*Sqrt[2])

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 250, normalized size = 2.94 \[ \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + {\left ({\left (\sqrt {3} - 2\right )} \cosh \relax (x) + {\left (\sqrt {3} - 2\right )} \sinh \relax (x)\right )} \sqrt {\sqrt {3} + 2} + 1\right ) - \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - {\left ({\left (\sqrt {3} - 2\right )} \cosh \relax (x) + {\left (\sqrt {3} - 2\right )} \sinh \relax (x)\right )} \sqrt {\sqrt {3} + 2} + 1\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + {\left ({\left (\sqrt {3} + 2\right )} \cosh \relax (x) + {\left (\sqrt {3} + 2\right )} \sinh \relax (x)\right )} \sqrt {-\sqrt {3} + 2} + 1\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - {\left ({\left (\sqrt {3} + 2\right )} \cosh \relax (x) + {\left (\sqrt {3} + 2\right )} \sinh \relax (x)\right )} \sqrt {-\sqrt {3} + 2} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) + 2}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x, algorithm="fricas")

[Out]

1/12*sqrt(sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(3) - 2)*cosh(x) + (sqrt(3) - 2)*
sinh(x))*sqrt(sqrt(3) + 2) + 1) - 1/12*sqrt(sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqr
t(3) - 2)*cosh(x) + (sqrt(3) - 2)*sinh(x))*sqrt(sqrt(3) + 2) + 1) - 1/12*sqrt(-sqrt(3) + 2)*log(cosh(x)^2 + 2*
cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)*sinh(x))*sqrt(-sqrt(3) + 2) + 1) + 1/12*s
qrt(-sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)*sinh(
x))*sqrt(-sqrt(3) + 2) + 1) + 1/12*sqrt(2)*log((cosh(x)^2 + sinh(x)^2 + 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + si
nh(x)^2))

________________________________________________________________________________________

giac [B]  time = 0.15, size = 154, normalized size = 1.81 \[ -\frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x, algorithm="giac")

[Out]

-1/24*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*e^x + e^(2*x) + 1) + 1/24*(sqrt(6) - sqrt(2))*log(-1/2*(
sqrt(6) + sqrt(2))*e^x + e^(2*x) + 1) - 1/24*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) - sqrt(2))*e^x + e^(2*x) + 1
) + 1/24*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*e^x + e^(2*x) + 1) + 1/12*sqrt(2)*log(sqrt(2)*e^x +
e^(2*x) + 1) - 1/12*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1)

________________________________________________________________________________________

maple [C]  time = 0.29, size = 78, normalized size = 0.92 \[ 2 \left (\munderset {\textit {\_R} =\RootOf \left (331776 \textit {\_Z}^{4}-2304 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (13824 \textit {\_R}^{3}-96 \textit {\_R} \right ) {\mathrm e}^{x}+1\right )\right )+\frac {\ln \left (1+{\mathrm e}^{2 x}+{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{12}-\frac {\ln \left (1+{\mathrm e}^{2 x}-{\mathrm e}^{x} \sqrt {2}\right ) \sqrt {2}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(6*x)*sinh(x),x)

[Out]

2*sum(_R*ln(exp(2*x)+(13824*_R^3-96*_R)*exp(x)+1),_R=RootOf(331776*_Z^4-2304*_Z^2+1))+1/12*ln(1+exp(2*x)+exp(x
)*2^(1/2))*2^(1/2)-1/12*ln(1+exp(2*x)-exp(x)*2^(1/2))*2^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \int \frac {e^{\left (7 \, x\right )} - e^{\left (5 \, x\right )} + e^{\left (3 \, x\right )} - e^{x}}{3 \, {\left (e^{\left (8 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x, algorithm="maxima")

[Out]

1/12*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) - 1/12*sqrt(2)*log(-sqrt(2)*e^x + e^(2*x) + 1) + integrate(1/3*(e^
(7*x) - e^(5*x) + e^(3*x) - e^x)/(e^(8*x) - e^(4*x) + 1), x)

________________________________________________________________________________________

mupad [B]  time = 1.72, size = 288, normalized size = 3.39 \[ \frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}-\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}+\ln \left (7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}-24\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+7\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-\ln \left (7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}+24\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+7\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+\ln \left (7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}-24\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+7\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}-\ln \left (7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}+24\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+7\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/cosh(6*x),x)

[Out]

(2^(1/2)*log(exp(2*x) + 2^(1/2)*exp(x) + 1))/12 - (2^(1/2)*log(exp(2*x) - 2^(1/2)*exp(x) + 1))/12 + log(7*exp(
2*x) - 4*3^(1/2) - 24*exp(x)*(1/72 - 3^(1/2)/144)^(1/2) - 4*3^(1/2)*exp(2*x) + 12*3^(1/2)*exp(x)*(1/72 - 3^(1/
2)/144)^(1/2) + 7)*(1/72 - 3^(1/2)/144)^(1/2) - log(7*exp(2*x) - 4*3^(1/2) + 24*exp(x)*(1/72 - 3^(1/2)/144)^(1
/2) - 4*3^(1/2)*exp(2*x) - 12*3^(1/2)*exp(x)*(1/72 - 3^(1/2)/144)^(1/2) + 7)*(1/72 - 3^(1/2)/144)^(1/2) + log(
7*exp(2*x) + 4*3^(1/2) - 24*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 4*3^(1/2)*exp(2*x) - 12*3^(1/2)*exp(x)*(3^(1/2
)/144 + 1/72)^(1/2) + 7)*(3^(1/2)/144 + 1/72)^(1/2) - log(7*exp(2*x) + 4*3^(1/2) + 24*exp(x)*(3^(1/2)/144 + 1/
72)^(1/2) + 4*3^(1/2)*exp(2*x) + 12*3^(1/2)*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 7)*(3^(1/2)/144 + 1/72)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (6 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(6*x)*sinh(x),x)

[Out]

Integral(sinh(x)*sech(6*x), x)

________________________________________________________________________________________