Integrand size = 8, antiderivative size = 289 \[ \int e^x \text {sech}(4 x) \, dx=\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} e^x}{1+e^{2 x}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} e^x}{1+e^{2 x}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \] Output:
1/2*arctan(((2-2^(1/2))^(1/2)-2*exp(x))/(2+2^(1/2))^(1/2))/(4+2*2^(1/2))^( 1/2)-1/2*arctan(((2+2^(1/2))^(1/2)-2*exp(x))/(2-2^(1/2))^(1/2))/(4-2*2^(1/ 2))^(1/2)-1/2*arctan(((2-2^(1/2))^(1/2)+2*exp(x))/(2+2^(1/2))^(1/2))/(4+2* 2^(1/2))^(1/2)+1/2*arctan(((2+2^(1/2))^(1/2)+2*exp(x))/(2-2^(1/2))^(1/2))/ (4-2*2^(1/2))^(1/2)+1/2*arctanh((2-2^(1/2))^(1/2)*exp(x)/(1+exp(2*x)))/(4- 2*2^(1/2))^(1/2)-1/2*arctanh((2+2^(1/2))^(1/2)*exp(x)/(1+exp(2*x)))/(4+2*2 ^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.08 \[ \int e^x \text {sech}(4 x) \, dx=\frac {2}{5} e^{5 x} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},1,\frac {13}{8},-e^{8 x}\right ) \] Input:
Integrate[E^x*Sech[4*x],x]
Output:
(2*E^(5*x)*Hypergeometric2F1[5/8, 1, 13/8, -E^(8*x)])/5
Time = 0.73 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {2720, 27, 828, 1447, 1475, 1083, 217, 1478, 25, 1103}
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.
| \(\displaystyle \int e^x \text {sech}(4 x) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int \frac {2 e^{4 x}}{e^{8 x}+1}de^x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^{4 x}}{1+e^{8 x}}de^x\) |
| \(\Big \downarrow \) 828 |
| \(\displaystyle 2 \left (\frac {\int \frac {e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}-\frac {\int \frac {e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1447 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \int \frac {1+e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x-\frac {1}{2} \int \frac {1-e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}-\frac {\frac {1}{2} \int \frac {1+e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x-\frac {1}{2} \int \frac {1-e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \int \frac {1}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x\right )-\frac {1}{2} \int \frac {1-e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}-\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \int \frac {1}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x\right )-\frac {1}{2} \int \frac {1-e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \left (-\int \frac {1}{-2+\sqrt {2}-e^{2 x}}d\left (-\sqrt {2+\sqrt {2}}+2 e^x\right )-\int \frac {1}{-2+\sqrt {2}-e^{2 x}}d\left (\sqrt {2+\sqrt {2}}+2 e^x\right )\right )-\frac {1}{2} \int \frac {1-e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}-\frac {\frac {1}{2} \left (-\int \frac {1}{-2-\sqrt {2}-e^{2 x}}d\left (-\sqrt {2-\sqrt {2}}+2 e^x\right )-\int \frac {1}{-2-\sqrt {2}-e^{2 x}}d\left (\sqrt {2-\sqrt {2}}+2 e^x\right )\right )-\frac {1}{2} \int \frac {1-e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \int \frac {1-e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}-\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \int \frac {1-e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1478 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\int -\frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2}}-\frac {\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\int -\frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}-\frac {\int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2}}-\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}-\frac {\int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2+\sqrt {2}}}-\frac {\log \left (\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2}}-\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2-\sqrt {2}}}-\frac {\log \left (\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2}}\right )\) |
Input:
Int[E^x*Sech[4*x],x]
Output:
2*(-1/2*((ArcTan[(-Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/Sqrt[2 + Sqrt[2]] + ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/Sqrt[2 + Sqrt[2]])/2 + (Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)]/(2*Sqrt[2 - Sqrt[2 ]]) - Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)]/(2*Sqrt[2 - Sqrt[2]]))/2)/S qrt[2] + ((ArcTan[(-Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/Sqrt[2 - Sqrt[2]] + ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/Sqrt[2 - Sqrt[2]])/2 + (Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]/(2*Sqrt[2 + Sqrt[ 2]]) - Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]/(2*Sqrt[2 + Sqrt[2]]))/2)/ (2*Sqrt[2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[R t[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[s^3/(2*Sqrt[2]*b*r) Int[x^ (m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Simp[s^3/(2*S qrt[2]*b*r) Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^ 2, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.76 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.09
| method | result | size |
| risch | \(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16777216 \textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left (-32768 \textit {\_R}^{5}+{\mathrm e}^{x}\right )\right )\) | \(25\) |
Input:
int(exp(x)*sech(4*x),x,method=_RETURNVERBOSE)
Output:
2*sum(_R*ln(-32768*_R^5+exp(x)),_R=RootOf(16777216*_Z^8+1))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.58 \[ \int e^x \text {sech}(4 x) \, dx=\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) - \frac {1}{4} \, \left (-1\right )^{\frac {1}{8}} \log \left (\left (-1\right )^{\frac {5}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{4} i \, \left (-1\right )^{\frac {1}{8}} \log \left (i \, \left (-1\right )^{\frac {5}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \frac {1}{4} i \, \left (-1\right )^{\frac {1}{8}} \log \left (-i \, \left (-1\right )^{\frac {5}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \frac {1}{4} \, \left (-1\right )^{\frac {1}{8}} \log \left (-\left (-1\right )^{\frac {5}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) \] Input:
integrate(exp(x)*sech(4*x),x, algorithm="fricas")
Output:
(1/8*I + 1/8)*sqrt(2)*(-1)^(1/8)*log((I + 1)*sqrt(2)*(-1)^(5/8) + 2*cosh(x ) + 2*sinh(x)) - (1/8*I - 1/8)*sqrt(2)*(-1)^(1/8)*log(-(I - 1)*sqrt(2)*(-1 )^(5/8) + 2*cosh(x) + 2*sinh(x)) + (1/8*I - 1/8)*sqrt(2)*(-1)^(1/8)*log((I - 1)*sqrt(2)*(-1)^(5/8) + 2*cosh(x) + 2*sinh(x)) - (1/8*I + 1/8)*sqrt(2)* (-1)^(1/8)*log(-(I + 1)*sqrt(2)*(-1)^(5/8) + 2*cosh(x) + 2*sinh(x)) - 1/4* (-1)^(1/8)*log((-1)^(5/8) + cosh(x) + sinh(x)) - 1/4*I*(-1)^(1/8)*log(I*(- 1)^(5/8) + cosh(x) + sinh(x)) + 1/4*I*(-1)^(1/8)*log(-I*(-1)^(5/8) + cosh( x) + sinh(x)) + 1/4*(-1)^(1/8)*log(-(-1)^(5/8) + cosh(x) + sinh(x))
\[ \int e^x \text {sech}(4 x) \, dx=\int e^{x} \operatorname {sech}{\left (4 x \right )}\, dx \] Input:
integrate(exp(x)*sech(4*x),x)
Output:
Integral(exp(x)*sech(4*x), x)
\[ \int e^x \text {sech}(4 x) \, dx=\int { e^{x} \operatorname {sech}\left (4 \, x\right ) \,d x } \] Input:
integrate(exp(x)*sech(4*x),x, algorithm="maxima")
Output:
integrate(e^x*sech(4*x), x)
Time = 0.16 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.86 \[ \int e^x \text {sech}(4 x) \, dx=\frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \] Input:
integrate(exp(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)) - 1/8*sqrt(-sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/8*sqrt(-sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*e^x + e^(2*x ) + 1) + 1/8*sqrt(sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) - 1/8*sqrt(sqrt(2) + 2)*log(-sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1)
Time = 5.84 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.66 \[ \int e^x \text {sech}(4 x) \, dx =\text {Too large to display} \] Input:
int(exp(x)/cosh(4*x),x)
Output:
log(32768*exp(x)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)^3 + 512)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8) - log(32768*exp( x)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)^3 - 512)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8) - log(32768*exp(x)*(((2^(1/2) + 2)^(1/2)*1i)/8 - (2 - 2^(1/2))^(1/2)/8)^3 - 512)*(((2^(1/2) + 2)^(1/2)*1 i)/8 - (2 - 2^(1/2))^(1/2)/8) + log(32768*exp(x)*(((2^(1/2) + 2)^(1/2)*1i) /8 - (2 - 2^(1/2))^(1/2)/8)^3 + 512)*(((2^(1/2) + 2)^(1/2)*1i)/8 - (2 - 2^ (1/2))^(1/2)/8) + 2^(1/2)*log(2^(1/2)*exp(x)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)^3*(16384 - 16384i) - 512)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 + 1i/2) - 2^(1/2)*log(2^(1/2)*exp(x)*((2 ^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)^3*(16384 - 16384i) + 512 )*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 + 1i/2) + 2^(1 /2)*log(2^(1/2)*exp(x)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8 )^3*(16384 + 16384i) - 512)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)* 1i)/8)*(1/2 - 1i/2) - 2^(1/2)*log(2^(1/2)*exp(x)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)^3*(16384 + 16384i) + 512)*((2^(1/2) + 2)^(1/2) /8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 - 1i/2)
Time = 0.26 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.62 \[ \int e^x \text {sech}(4 x) \, dx=\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}+\frac {\sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}+\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}+\frac {\sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )}{4}-\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}-\frac {\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8}-\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8}-\frac {\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8} \] Input:
int(exp(x)*sech(4*x),x)
Output:
(2*sqrt(sqrt(2) + 2)*sqrt(2)*atan((sqrt( - sqrt(2) + 2) - 2*e**x)/sqrt(sqr t(2) + 2)) - 2*sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) - 2*e**x)/sqrt (sqrt(2) + 2)) - 2*sqrt(sqrt(2) + 2)*sqrt(2)*atan((sqrt( - sqrt(2) + 2) + 2*e**x)/sqrt(sqrt(2) + 2)) + 2*sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2 ) + 2*e**x)/sqrt(sqrt(2) + 2)) - 2*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((sqrt (sqrt(2) + 2) - 2*e**x)/sqrt( - sqrt(2) + 2)) - 2*sqrt( - sqrt(2) + 2)*ata n((sqrt(sqrt(2) + 2) - 2*e**x)/sqrt( - sqrt(2) + 2)) + 2*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((sqrt(sqrt(2) + 2) + 2*e**x)/sqrt( - sqrt(2) + 2)) + 2*sq rt( - sqrt(2) + 2)*atan((sqrt(sqrt(2) + 2) + 2*e**x)/sqrt( - sqrt(2) + 2)) - sqrt( - sqrt(2) + 2)*sqrt(2)*log( - e**x*sqrt( - sqrt(2) + 2) + e**(2*x ) + 1) + sqrt( - sqrt(2) + 2)*sqrt(2)*log(e**x*sqrt( - sqrt(2) + 2) + e**( 2*x) + 1) - sqrt( - sqrt(2) + 2)*log( - e**x*sqrt( - sqrt(2) + 2) + e**(2* x) + 1) + sqrt( - sqrt(2) + 2)*log(e**x*sqrt( - sqrt(2) + 2) + e**(2*x) + 1) + sqrt(sqrt(2) + 2)*sqrt(2)*log( - e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) - sqrt(sqrt(2) + 2)*sqrt(2)*log(e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) - sqrt(sqrt(2) + 2)*log( - e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) + sqrt(s qrt(2) + 2)*log(e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1))/8