\(\int \cosh (x) \text {sech}(6 x) \, dx\) [126]

Optimal result

Integrand size = 7, antiderivative size = 85 \[ \int \cosh (x) \text {sech}(6 x) \, dx=-\frac {\arctan \left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}+\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \] Output:

-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))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \cosh (x) \text {sech}(6 x) \, dx=\frac {1}{6} \left (-\sqrt {2} \arctan \left (\sqrt {2} \sinh (x)\right )+\sqrt {2+\sqrt {3}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )+\sqrt {2-\sqrt {3}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )\right ) \] Input:

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

Output:

(-(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
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4856, 2460, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (i x)}{\cos (6 i x)}dx\)

\(\Big \downarrow \) 4856

\(\displaystyle \int \frac {1}{32 \sinh ^6(x)+48 \sinh ^4(x)+18 \sinh ^2(x)+1}d\sinh (x)\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (\frac {4 \left (2 \sinh ^2(x)+1\right )}{3 \left (16 \sinh ^4(x)+16 \sinh ^2(x)+1\right )}-\frac {1}{3 \left (2 \sinh ^2(x)+1\right )}\right )d\sinh (x)\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}+\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}}\)

Input:

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

Output:

-1/3*ArcTan[Sqrt[2]*Sinh[x]]/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]])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2460

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, 
Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /;  !SumQ[NonfreeFactors[Q 
x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] &&  !BinomialQ[Px, x] && 
 !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4856

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto 
rs[Sin[c*(a + b*x)], x]}, Simp[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])
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98

method	result	size
risch	\(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (331776 \textit {\_Z}^{4}+2304 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (13824 \textit {\_R}^{3}+96 \textit {\_R} \right ) {\mathrm e}^{x}-1\right )\right )+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{12}\)	\(83\)

Input:

int(cosh(x)*sech(6*x),x,method=_RETURNVERBOSE)
 

Output:

2*sum(_R*ln(exp(2*x)+(13824*_R^3+96*_R)*exp(x)-1),_R=RootOf(331776*_Z^4+23 
04*_Z^2+1))+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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (67) = 134\).

Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.81 \[ \int \cosh (x) \text {sech}(6 x) \, dx=\frac {1}{6} \, \sqrt {\sqrt {3} + 2} \arctan \left ({\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} - {\left (\sqrt {3} - 1\right )} \cosh \left (x\right ) + {\left (3 \, \cosh \left (x\right )^{2} - \sqrt {3} + 1\right )} \sinh \left (x\right )\right )} \sqrt {\sqrt {3} + 2}\right ) + \frac {1}{6} \, \sqrt {-\sqrt {3} + 2} \arctan \left ({\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (\sqrt {3} + 1\right )} \cosh \left (x\right ) + {\left (3 \, \cosh \left (x\right )^{2} + \sqrt {3} + 1\right )} \sinh \left (x\right )\right )} \sqrt {-\sqrt {3} + 2}\right ) + \frac {1}{6} \, \sqrt {\sqrt {3} + 2} \arctan \left (\sqrt {\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}\right ) + \frac {1}{6} \, \sqrt {-\sqrt {3} + 2} \arctan \left (\sqrt {-\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \cosh \left (x\right ) + \frac {1}{2} \, \sqrt {2} \sinh \left (x\right )\right ) + \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}}{2 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) \] Input:

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

Output:

1/6*sqrt(sqrt(3) + 2)*arctan((cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 
- (sqrt(3) - 1)*cosh(x) + (3*cosh(x)^2 - sqrt(3) + 1)*sinh(x))*sqrt(sqrt(3 
) + 2)) + 1/6*sqrt(-sqrt(3) + 2)*arctan((cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + 
 sinh(x)^3 + (sqrt(3) + 1)*cosh(x) + (3*cosh(x)^2 + sqrt(3) + 1)*sinh(x))* 
sqrt(-sqrt(3) + 2)) + 1/6*sqrt(sqrt(3) + 2)*arctan(sqrt(sqrt(3) + 2)*(cosh 
(x) + sinh(x))) + 1/6*sqrt(-sqrt(3) + 2)*arctan(sqrt(-sqrt(3) + 2)*(cosh(x 
) + sinh(x))) - 1/6*sqrt(2)*arctan(1/2*sqrt(2)*cosh(x) + 1/2*sqrt(2)*sinh( 
x)) + 1/6*sqrt(2)*arctan(-1/2*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh( 
x) + sqrt(2)*sinh(x)^2 + sqrt(2))/(cosh(x) - sinh(x)))
 

Sympy [F]

\[ \int \cosh (x) \text {sech}(6 x) \, dx=\int \cosh {\left (x \right )} \operatorname {sech}{\left (6 x \right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \cosh (x) \text {sech}(6 x) \, dx=\int { \cosh \left (x\right ) \operatorname {sech}\left (6 \, x\right ) \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (67) = 134\).

Time = 0.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.08 \[ \int \cosh (x) \text {sech}(6 x) \, dx=\frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, e^{x}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, e^{x}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, e^{x}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, e^{x}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) \] Input:

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

Output:

1/12*(sqrt(6) - sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*e^x)/(sqrt(6) + sqr 
t(2))) + 1/12*(sqrt(6) - sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*e^x)/(sqr 
t(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) + sq 
rt(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))
 

Mupad [B] (verification not implemented)

Time = 3.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.42 \[ \int \cosh (x) \text {sech}(6 x) \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}-\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}-\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}+\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-7}{\frac {5\,\sqrt {2}\,{\mathrm {e}}^x}{2}+\frac {3\,\sqrt {6}\,{\mathrm {e}}^x}{2}}\right )}{12}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{2}\right )}{6} \] Input:

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

Output:

(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*ex 
p(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
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.38 \[ \int \cosh (x) \text {sech}(6 x) \, dx=\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {4 e^{x}-\sqrt {6}-\sqrt {2}}{2 \sqrt {-\sqrt {3}+2}}\right )}{6}+\frac {\sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {4 e^{x}-\sqrt {6}-\sqrt {2}}{2 \sqrt {-\sqrt {3}+2}}\right )}{3}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {4 e^{x}+\sqrt {6}+\sqrt {2}}{2 \sqrt {-\sqrt {3}+2}}\right )}{6}+\frac {\sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {4 e^{x}+\sqrt {6}+\sqrt {2}}{2 \sqrt {-\sqrt {3}+2}}\right )}{3}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}-\sqrt {2}}{\sqrt {2}}\right )}{6}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}+\sqrt {2}}{\sqrt {2}}\right )}{6}-\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )}{12}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )}{12}+\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )}{12}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )}{12} \] Input:

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

Output:

(2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((4*e**x - sqrt(6) - sqrt(2))/(2*sqrt( 
 - sqrt(3) + 2))) + 4*sqrt( - sqrt(3) + 2)*atan((4*e**x - sqrt(6) - sqrt(2 
))/(2*sqrt( - sqrt(3) + 2))) + 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((4*e**x 
 + sqrt(6) + sqrt(2))/(2*sqrt( - sqrt(3) + 2))) + 4*sqrt( - sqrt(3) + 2)*a 
tan((4*e**x + sqrt(6) + sqrt(2))/(2*sqrt( - sqrt(3) + 2))) - 2*sqrt(2)*ata 
n((2*e**x - sqrt(2))/sqrt(2)) - 2*sqrt(2)*atan((2*e**x + sqrt(2))/sqrt(2)) 
 - sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4*e**x)/(sqrt(6) + sqrt(2))) + s 
qrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4*e**x)/(sqrt(6) + sqrt(2))) + sqrt( 
6)*atan((2*sqrt( - sqrt(3) + 2) + 4*e**x)/(sqrt(6) + sqrt(2))) - sqrt(2)*a 
tan((2*sqrt( - sqrt(3) + 2) + 4*e**x)/(sqrt(6) + sqrt(2))))/12