### 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]

-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]])

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Rubi [A]  time = 0.0576444, antiderivative size = 85, normalized size of antiderivative = 1., 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 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])

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 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]

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 = 0.0843409, 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

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Maple [C]  time = 0.083, size = 83, normalized size = 1. \begin{align*}{\frac{i}{12}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{2}{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{2}{{\rm e}^{x}}-1 \right ) +2\,\sum _{{\it \_R}={\it RootOf} \left ( 331776\,{{\it \_Z}}^{4}+2304\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{2\,x}}+ \left ( 13824\,{{\it \_R}}^{3}+96\,{\it \_R} \right ){{\rm e}^{x}}-1 \right ) \end{align*}

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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}

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)

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Fricas [B]  time = 2.38837, size = 489, normalized size = 5.75 \begin{align*} -\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 ) \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \operatorname{sech}{\left (6 x \right )}\, dx \end{align*}

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)

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Giac [B]  time = 1.20785, size = 239, normalized size = 2.81 \begin{align*} \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 ) \end{align*}

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))