Integrand size = 7, antiderivative size = 71 \[ \int \cosh (x) \text {sech}(4 x) \, dx=\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \] Output:
1/2*arctan(2*sinh(x)/(2-2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/2*arctan(2*s inh(x)/(2+2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \cosh (x) \text {sech}(4 x) \, dx=\frac {1}{4} \sqrt {2+\sqrt {2}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \] Input:
Integrate[Cosh[x]*Sech[4*x],x]
Output:
(Sqrt[2 + Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]])/4 - ArcTan[(2*Si nh[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])
Time = 0.25 (sec) , antiderivative size = 71, 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, 1406, 216}
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.
\(\displaystyle \int \cosh (x) \text {sech}(4 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i x)}{\cos (4 i x)}dx\) |
\(\Big \downarrow \) 4856 |
\(\displaystyle \int \frac {1}{8 \sinh ^4(x)+8 \sinh ^2(x)+1}d\sinh (x)\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \sqrt {2} \int \frac {1}{8 \sinh ^2(x)+2 \left (2-\sqrt {2}\right )}d\sinh (x)-\sqrt {2} \int \frac {1}{8 \sinh ^2(x)+2 \left (2+\sqrt {2}\right )}d\sinh (x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}\) |
Input:
Int[Cosh[x]*Sech[4*x],x]
Output:
ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[( 2*Sinh[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[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]
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])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56
method | result | size |
risch | \(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (32768 \textit {\_Z}^{4}+512 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-4096 \textit {\_R}^{3}-48 \textit {\_R} \right ) {\mathrm e}^{x}-1\right )\right )\) | \(40\) |
Input:
int(cosh(x)*sech(4*x),x,method=_RETURNVERBOSE)
Output:
2*sum(_R*ln(exp(2*x)+(-4096*_R^3-48*_R)*exp(x)-1),_R=RootOf(32768*_Z^4+512 *_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (49) = 98\).
Time = 0.10 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.06 \[ \int \cosh (x) \text {sech}(4 x) \, dx=\frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (\sqrt {2} - 2\right )} \cosh \left (x\right ) + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2} - 2\right )} \sinh \left (x\right )\right )} \sqrt {\sqrt {2} + 2}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\sqrt {2} + 2}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (\sqrt {2} + 2\right )} \cosh \left (x\right ) + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2} + 2\right )} \sinh \left (x\right )\right )} \sqrt {-\sqrt {2} + 2}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {-\sqrt {2} + 2}\right ) \] Input:
integrate(cosh(x)*sech(4*x),x, algorithm="fricas")
Output:
1/4*sqrt(sqrt(2) + 2)*arctan(1/2*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*si nh(x)^2 + sqrt(2)*sinh(x)^3 + (sqrt(2) - 2)*cosh(x) + (3*sqrt(2)*cosh(x)^2 + sqrt(2) - 2)*sinh(x))*sqrt(sqrt(2) + 2)) + 1/4*sqrt(sqrt(2) + 2)*arctan (1/2*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sqrt(2) + 2)) - 1/4*sqrt(-sq rt(2) + 2)*arctan(1/2*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + s qrt(2)*sinh(x)^3 + (sqrt(2) + 2)*cosh(x) + (3*sqrt(2)*cosh(x)^2 + sqrt(2) + 2)*sinh(x))*sqrt(-sqrt(2) + 2)) - 1/4*sqrt(-sqrt(2) + 2)*arctan(1/2*(sqr t(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(-sqrt(2) + 2))
\[ \int \cosh (x) \text {sech}(4 x) \, dx=\int \cosh {\left (x \right )} \operatorname {sech}{\left (4 x \right )}\, dx \] Input:
integrate(cosh(x)*sech(4*x),x)
Output:
Integral(cosh(x)*sech(4*x), x)
\[ \int \cosh (x) \text {sech}(4 x) \, dx=\int { \cosh \left (x\right ) \operatorname {sech}\left (4 \, x\right ) \,d x } \] Input:
integrate(cosh(x)*sech(4*x),x, algorithm="maxima")
Output:
integrate(cosh(x)*sech(4*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (49) = 98\).
Time = 0.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.90 \[ \int \cosh (x) \text {sech}(4 x) \, dx=\frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) \] Input:
integrate(cosh(x)*sech(4*x),x, algorithm="giac")
Output:
1/4*sqrt(sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*e^x)/sqrt(-sqrt(2) + 2 )) + 1/4*sqrt(sqrt(2) + 2)*arctan(-(sqrt(sqrt(2) + 2) - 2*e^x)/sqrt(-sqrt( 2) + 2)) - 1/4*sqrt(-sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*e^x)/sqrt (sqrt(2) + 2)) - 1/4*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*e^ x)/sqrt(sqrt(2) + 2))
Time = 2.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.77 \[ \int \cosh (x) \text {sech}(4 x) \, dx=\frac {\mathrm {atan}\left (\frac {3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3}{{\mathrm {e}}^x\,\sqrt {\sqrt {2}+2}+\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\sqrt {2}+2}}\right )\,\sqrt {\sqrt {2}+2}}{4}+\frac {\mathrm {atan}\left (\frac {3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3}{{\mathrm {e}}^x\,\sqrt {2-\sqrt {2}}-\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {2-\sqrt {2}}}\right )\,\sqrt {2-\sqrt {2}}}{4} \] Input:
int(cosh(x)/cosh(4*x),x)
Output:
(atan((3*exp(2*x) - 2*2^(1/2) + 2*2^(1/2)*exp(2*x) - 3)/(exp(x)*(2^(1/2) + 2)^(1/2) + 2^(1/2)*exp(x)*(2^(1/2) + 2)^(1/2)))*(2^(1/2) + 2)^(1/2))/4 + (atan((3*exp(2*x) + 2*2^(1/2) - 2*2^(1/2)*exp(2*x) - 3)/(exp(x)*(2 - 2^(1/ 2))^(1/2) - 2^(1/2)*exp(x)*(2 - 2^(1/2))^(1/2)))*(2 - 2^(1/2))^(1/2))/4
Time = 0.24 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.51 \[ \int \cosh (x) \text {sech}(4 x) \, dx=\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}+\frac {\sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}+\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}+\frac {\sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4} \] Input:
int(cosh(x)*sech(4*x),x)
Output:
(sqrt(sqrt(2) + 2)*sqrt(2)*atan((sqrt( - sqrt(2) + 2) - 2*e**x)/sqrt(sqrt( 2) + 2)) - sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) - 2*e**x)/sqrt(sqr t(2) + 2)) - sqrt(sqrt(2) + 2)*sqrt(2)*atan((sqrt( - sqrt(2) + 2) + 2*e**x )/sqrt(sqrt(2) + 2)) + sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) + 2*e* *x)/sqrt(sqrt(2) + 2)) - sqrt( - sqrt(2) + 2)*sqrt(2)*atan((sqrt(sqrt(2) + 2) - 2*e**x)/sqrt( - sqrt(2) + 2)) - sqrt( - sqrt(2) + 2)*atan((sqrt(sqrt (2) + 2) - 2*e**x)/sqrt( - sqrt(2) + 2)) + sqrt( - sqrt(2) + 2)*sqrt(2)*at an((sqrt(sqrt(2) + 2) + 2*e**x)/sqrt( - sqrt(2) + 2)) + sqrt( - sqrt(2) + 2)*atan((sqrt(sqrt(2) + 2) + 2*e**x)/sqrt( - sqrt(2) + 2)))/4