3.244 \(\int \cosh (x) \text {sech}(6 x) \, dx\)

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

[Out]

-1/6*arctan(sinh(x)*2^(1/2))*2^(1/2)+1/6*arctan(2*sinh(x)/(1/2*6^(1/2)-1/2*2^(1/2)))/(1/2*6^(1/2)-1/2*2^(1/2))
+1/6*arctan(2*sinh(x)/(1/2*6^(1/2)+1/2*2^(1/2)))/(1/2*6^(1/2)+1/2*2^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.06, 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 = {4356, 2057, 203, 1166} \[ -\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[Sqrt[2]*Sinh[x]]/(3*Sqrt[2]) + ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTan[(2
*Sinh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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 4356

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

Rubi steps

\begin {align*} \int \cosh (x) \text {sech}(6 x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+18 x^2+48 x^4+32 x^6} \, dx,x,\sinh (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,\sinh (x)\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\sinh (x)\right )\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1+2 x^2}{1+16 x^2+16 x^4} \, dx,x,\sinh (x)\right )\\ &=-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{8-4 \sqrt {3}+16 x^2} \, dx,x,\sinh (x)\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{8+4 \sqrt {3}+16 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {\tan ^{-1}\left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 81, normalized size = 0.95 \[ \frac {1}{6} \left (-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sinh (x)\right )+\sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )+\sqrt {2-\sqrt {3}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2]*ArcTan[Sqrt[2]*Sinh[x]]) + Sqrt[2 + Sqrt[3]]*ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[3]]] + Sqrt[2 - Sqrt[
3]]*ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[3]]])/6

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 156, normalized size = 1.84 \[ -\frac {1}{3} \, \sqrt {\sqrt {3} + 2} \arctan \left (-{\left (\sqrt {\sqrt {3} + 2} {\left (e^{\left (2 \, x\right )} - 1\right )} - \sqrt {-\sqrt {3} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt {\sqrt {3} + 2}\right )} e^{\left (-x\right )}\right ) - \frac {1}{3} \, \sqrt {-\sqrt {3} + 2} \arctan \left (-{\left (\sqrt {-\sqrt {3} + 2} {\left (e^{\left (2 \, x\right )} - 1\right )} - \sqrt {\sqrt {3} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt {-\sqrt {3} + 2}\right )} e^{\left (-x\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} e^{\left (3 \, x\right )} + \frac {1}{2} \, \sqrt {2} e^{x}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(sqrt(3) + 2)*arctan(-(sqrt(sqrt(3) + 2)*(e^(2*x) - 1) - sqrt(-sqrt(3)*e^(2*x) + e^(4*x) + 1)*sqrt(sq
rt(3) + 2))*e^(-x)) - 1/3*sqrt(-sqrt(3) + 2)*arctan(-(sqrt(-sqrt(3) + 2)*(e^(2*x) - 1) - sqrt(sqrt(3)*e^(2*x)
+ e^(4*x) + 1)*sqrt(-sqrt(3) + 2))*e^(-x)) - 1/6*sqrt(2)*arctan(1/2*sqrt(2)*e^(3*x) + 1/2*sqrt(2)*e^x) - 1/6*s
qrt(2)*arctan(1/2*sqrt(2)*e^x)

________________________________________________________________________________________

giac [B]  time = 0.13, size = 177, normalized size = 2.08 \[ \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, e^{x}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, e^{x}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, e^{x}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, e^{x}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(sqrt(6) - sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*e^x)/(sqrt(6) + sqrt(2))) + 1/12*(sqrt(6) - sqrt(2))*ar
ctan(-(sqrt(6) - sqrt(2) - 4*e^x)/(sqrt(6) + sqrt(2))) + 1/12*(sqrt(6) + sqrt(2))*arctan((sqrt(6) + sqrt(2) +
4*e^x)/(sqrt(6) - sqrt(2))) + 1/12*(sqrt(6) + sqrt(2))*arctan(-(sqrt(6) + sqrt(2) - 4*e^x)/(sqrt(6) - sqrt(2))
) - 1/6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) - 1/6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x))

________________________________________________________________________________________

maple [C]  time = 0.32, size = 83, normalized size = 0.98 \[ \frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{12}+2 \left (\munderset {\textit {\_R} =\RootOf \left (331776 \textit {\_Z}^{4}+2304 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (13824 \textit {\_R}^{3}+96 \textit {\_R} \right ) {\mathrm e}^{x}-1\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) + \int \frac {e^{\left (7 \, x\right )} + e^{\left (5 \, x\right )} + e^{\left (3 \, x\right )} + e^{x}}{3 \, {\left (e^{\left (8 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x)) - 1/6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) + inte
grate(1/3*(e^(7*x) + e^(5*x) + e^(3*x) + e^x)/(e^(8*x) - e^(4*x) + 1), x)

________________________________________________________________________________________

mupad [B]  time = 3.53, size = 206, normalized size = 2.42 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}-\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}-\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{2}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(1/2)*atan((7*exp(2*x) + 4*3^(1/2) - 4*3^(1/2)*exp(2*x) - 7)/((5*2^(1/2)*exp(x))/2 - (3*6^(1/2)*exp(x))/2))
)/12 + (2^(1/2)*atan((7*exp(2*x) - 4*3^(1/2) + 4*3^(1/2)*exp(2*x) - 7)/((5*2^(1/2)*exp(x))/2 + (3*6^(1/2)*exp(
x))/2)))/12 - (6^(1/2)*atan((7*exp(2*x) + 4*3^(1/2) - 4*3^(1/2)*exp(2*x) - 7)/((5*2^(1/2)*exp(x))/2 - (3*6^(1/
2)*exp(x))/2)))/12 + (6^(1/2)*atan((7*exp(2*x) - 4*3^(1/2) + 4*3^(1/2)*exp(2*x) - 7)/((5*2^(1/2)*exp(x))/2 + (
3*6^(1/2)*exp(x))/2)))/12 - (2^(1/2)*atan((2^(1/2)*exp(-x)*(exp(2*x) - 1))/2))/6

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)*sech(6*x), x)

________________________________________________________________________________________