Integrand size = 10, antiderivative size = 293 \[ \int e^x \text {sech}^2(4 x) \, dx=-\frac {e^x}{2 \left (1+e^{8 x}\right )}-\frac {1}{16} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 e^x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{16} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 e^x}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{16} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} e^x}{1+e^{2 x}}\right )+\frac {1}{16} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} e^x}{1+e^{2 x}}\right ) \] Output:
-1/2*exp(x)/(1+exp(8*x))-1/16*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1/2)- 2*exp(x))/(2+2^(1/2))^(1/2))-1/16*(2-2^(1/2))^(1/2)*arctan(((2+2^(1/2))^(1 /2)-2*exp(x))/(2-2^(1/2))^(1/2))+1/16*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2) )^(1/2)+2*exp(x))/(2+2^(1/2))^(1/2))+1/16*(2-2^(1/2))^(1/2)*arctan(((2+2^( 1/2))^(1/2)+2*exp(x))/(2-2^(1/2))^(1/2))+1/16*(2-2^(1/2))^(1/2)*arctanh((2 -2^(1/2))^(1/2)*exp(x)/(1+exp(2*x)))+1/16*(2+2^(1/2))^(1/2)*arctanh((2+2^( 1/2))^(1/2)*exp(x)/(1+exp(2*x)))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.12 \[ \int e^x \text {sech}^2(4 x) \, dx=\frac {1}{2} e^x \left (-\frac {1}{1+e^{8 x}}+\operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{8 x}\right )\right ) \] Input:
Integrate[E^x*Sech[4*x]^2,x]
Output:
(E^x*(-(1 + E^(8*x))^(-1) + Hypergeometric2F1[1/8, 1, 9/8, -E^(8*x)]))/2
Time = 0.75 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2720, 27, 817, 757, 1483, 1142, 25, 1083, 217, 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.
| \(\displaystyle \int e^x \text {sech}^2(4 x) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int \frac {4 e^{8 x}}{\left (e^{8 x}+1\right )^2}de^x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {e^{8 x}}{\left (1+e^{8 x}\right )^2}de^x\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 4 \left (\frac {1}{8} \int \frac {1}{1+e^{8 x}}de^x-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 757 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\int \frac {\sqrt {2}-e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}+e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}+\left (1-\sqrt {2}\right ) e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}-\left (1-\sqrt {2}\right ) e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}-\left (1+\sqrt {2}\right ) e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}+\left (1+\sqrt {2}\right ) e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\frac {-\sqrt {2+\sqrt {2}} \int \frac {1}{-2-\sqrt {2}-e^{2 x}}d\left (-\sqrt {2-\sqrt {2}}+2 e^x\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {-\sqrt {2+\sqrt {2}} \int \frac {1}{-2-\sqrt {2}-e^{2 x}}d\left (\sqrt {2-\sqrt {2}}+2 e^x\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x-\sqrt {2-\sqrt {2}} \int \frac {1}{-2+\sqrt {2}-e^{2 x}}d\left (-\sqrt {2+\sqrt {2}}+2 e^x\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x-\sqrt {2-\sqrt {2}} \int \frac {1}{-2+\sqrt {2}-e^{2 x}}d\left (\sqrt {2+\sqrt {2}}+2 e^x\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 4 \left (\frac {1}{8} \left (\frac {\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (-\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (-\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )-\frac {e^x}{8 \left (e^{8 x}+1\right )}\right )\) |
Input:
Int[E^x*Sech[4*x]^2,x]
Output:
4*(-1/8*E^x/(1 + E^(8*x)) + (((ArcTan[(-Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)])/2)/ (2*Sqrt[2 - Sqrt[2]]) + (ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[ 2]]] - ((1 - Sqrt[2])*Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)])/2)/(2*Sqrt [2 - Sqrt[2]]))/(2*Sqrt[2]) + ((ArcTan[(-Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]] - ((1 + Sqrt[2])*Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)])/2) /(2*Sqrt[2 + Sqrt[2]]) + (ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt [2]]] + ((1 + Sqrt[2])*Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)])/2)/(2*Sqr t[2 + Sqrt[2]]))/(2*Sqrt[2]))/8)
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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[r/(2*Sqrt[2]*a) Int[(Sqrt[2]*r - s*x^(n/4))/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] + Simp[r/(2*S qrt[2]*a) Int[(Sqrt[2]*r + s*x^(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, 1] && GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
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 = 1.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.12
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{x}}{2 \left (1+{\mathrm e}^{8 x}\right )}+4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (281474976710656 \textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+64 \textit {\_R} \right )\right )\) | \(36\) |
| default | \(\frac {-\frac {3 \tanh \left (\frac {x}{2}\right )^{7}}{2}-\frac {7 \tanh \left (\frac {x}{2}\right )^{6}}{2}-\frac {7 \tanh \left (\frac {x}{2}\right )^{5}}{2}-\frac {35 \tanh \left (\frac {x}{2}\right )^{4}}{2}+\frac {7 \tanh \left (\frac {x}{2}\right )^{3}}{2}-\frac {21 \tanh \left (\frac {x}{2}\right )^{2}}{2}+\frac {3 \tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{2}}{\tanh \left (\frac {x}{2}\right )^{8}+28 \tanh \left (\frac {x}{2}\right )^{6}+70 \tanh \left (\frac {x}{2}\right )^{4}+28 \tanh \left (\frac {x}{2}\right )^{2}+1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+28 \textit {\_Z}^{6}+70 \textit {\_Z}^{4}+28 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-6 \textit {\_R}^{5}+15 \textit {\_R}^{4}-20 \textit {\_R}^{3}+15 \textit {\_R}^{2}-6 \textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+21 \textit {\_R}^{5}+35 \textit {\_R}^{3}+7 \textit {\_R}}\right )}{16}\) | \(177\) |
Input:
int(exp(x)*sech(4*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*exp(x)/(1+exp(8*x))+4*sum(_R*ln(exp(x)+64*_R),_R=RootOf(2814749767106 56*_Z^8+1))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 1215, normalized size of antiderivative = 4.15 \[ \int e^x \text {sech}^2(4 x) \, dx=\text {Too large to display} \] Input:
integrate(exp(x)*sech(4*x)^2,x, algorithm="fricas")
Output:
1/32*(((I + 1)*sqrt(2)*(-1)^(1/8)*cosh(x)^8 + (8*I + 8)*sqrt(2)*(-1)^(1/8) *cosh(x)^7*sinh(x) + (28*I + 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^6*sinh(x)^2 + (56*I + 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^5*sinh(x)^3 + (70*I + 70)*sqrt(2)*( -1)^(1/8)*cosh(x)^4*sinh(x)^4 + (56*I + 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^3*s inh(x)^5 + (28*I + 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^2*sinh(x)^6 + (8*I + 8)* sqrt(2)*(-1)^(1/8)*cosh(x)*sinh(x)^7 + (I + 1)*sqrt(2)*(-1)^(1/8)*sinh(x)^ 8 + (I + 1)*sqrt(2)*(-1)^(1/8))*log((I + 1)*sqrt(2)*(-1)^(1/8) + 2*cosh(x) + 2*sinh(x)) + (-(I - 1)*sqrt(2)*(-1)^(1/8)*cosh(x)^8 - (8*I - 8)*sqrt(2) *(-1)^(1/8)*cosh(x)^7*sinh(x) - (28*I - 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^6*s inh(x)^2 - (56*I - 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^5*sinh(x)^3 - (70*I - 70 )*sqrt(2)*(-1)^(1/8)*cosh(x)^4*sinh(x)^4 - (56*I - 56)*sqrt(2)*(-1)^(1/8)* cosh(x)^3*sinh(x)^5 - (28*I - 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^2*sinh(x)^6 - (8*I - 8)*sqrt(2)*(-1)^(1/8)*cosh(x)*sinh(x)^7 - (I - 1)*sqrt(2)*(-1)^(1/ 8)*sinh(x)^8 - (I - 1)*sqrt(2)*(-1)^(1/8))*log(-(I - 1)*sqrt(2)*(-1)^(1/8) + 2*cosh(x) + 2*sinh(x)) + ((I - 1)*sqrt(2)*(-1)^(1/8)*cosh(x)^8 + (8*I - 8)*sqrt(2)*(-1)^(1/8)*cosh(x)^7*sinh(x) + (28*I - 28)*sqrt(2)*(-1)^(1/8)* cosh(x)^6*sinh(x)^2 + (56*I - 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^5*sinh(x)^3 + (70*I - 70)*sqrt(2)*(-1)^(1/8)*cosh(x)^4*sinh(x)^4 + (56*I - 56)*sqrt(2)* (-1)^(1/8)*cosh(x)^3*sinh(x)^5 + (28*I - 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^2* sinh(x)^6 + (8*I - 8)*sqrt(2)*(-1)^(1/8)*cosh(x)*sinh(x)^7 + (I - 1)*sq...
\[ \int e^x \text {sech}^2(4 x) \, dx=\int e^{x} \operatorname {sech}^{2}{\left (4 x \right )}\, dx \] Input:
integrate(exp(x)*sech(4*x)**2,x)
Output:
Integral(exp(x)*sech(4*x)**2, x)
\[ \int e^x \text {sech}^2(4 x) \, dx=\int { e^{x} \operatorname {sech}\left (4 \, x\right )^{2} \,d x } \] Input:
integrate(exp(x)*sech(4*x)^2,x, algorithm="maxima")
Output:
-1/2*e^x/(e^(8*x) + 1) + 4*integrate(1/8*e^x/(e^(8*x) + 1), x)
Time = 0.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.89 \[ \int e^x \text {sech}^2(4 x) \, dx=\frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{32} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{32} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{32} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{32} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {e^{x}}{2 \, {\left (e^{\left (8 \, x\right )} + 1\right )}} \] Input:
integrate(exp(x)*sech(4*x)^2,x, algorithm="giac")
Output:
1/16*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*e^x)/sqrt(-sqrt(2) + 2)) + 1/16*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(sqrt(2) + 2) - 2*e^x)/sqrt(-s qrt(2) + 2)) + 1/16*sqrt(sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*e^x)/ sqrt(sqrt(2) + 2)) + 1/16*sqrt(sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*e^x)/sqrt(sqrt(2) + 2)) + 1/32*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*e ^x + e^(2*x) + 1) - 1/32*sqrt(sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*e^x + e^ (2*x) + 1) + 1/32*sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) - 1/32*sqrt(-sqrt(2) + 2)*log(-sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) - 1/2*e^x/(e^(8*x) + 1)
Time = 4.75 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.61 \[ \int e^x \text {sech}^2(4 x) \, dx =\text {Too large to display} \] Input:
int(exp(x)/cosh(4*x)^2,x)
Output:
log(- exp(x)/2 - (2^(1/2) + 2)^(1/2)/4 - ((2 - 2^(1/2))^(1/2)*1i)/4)*((2^( 1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32) - exp(x)/(2*(exp(8*x) + 1)) - log((2^(1/2) + 2)^(1/2)/4 - exp(x)/2 + ((2 - 2^(1/2))^(1/2)*1i)/4)*( (2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32) + log((2 - 2^(1/2))^ (1/2)/4 - ((2^(1/2) + 2)^(1/2)*1i)/4 - exp(x)/2)*(((2^(1/2) + 2)^(1/2)*1i) /32 - (2 - 2^(1/2))^(1/2)/32) - log(((2^(1/2) + 2)^(1/2)*1i)/4 - exp(x)/2 - (2 - 2^(1/2))^(1/2)/4)*(((2^(1/2) + 2)^(1/2)*1i)/32 - (2 - 2^(1/2))^(1/2 )/32) + 2^(1/2)*log(- exp(x)/2 - 2^(1/2)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2 ^(1/2))^(1/2)*1i)/32)*(4 + 4i))*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^( 1/2)*1i)/32)*(1/2 + 1i/2) + 2^(1/2)*log(- exp(x)/2 - 2^(1/2)*((2^(1/2) + 2 )^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(4 - 4i))*((2^(1/2) + 2)^(1/2)/3 2 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(1/2 - 1i/2) - 2^(1/2)*log(2^(1/2)*((2^(1 /2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(4 - 4i) - exp(x)/2)*((2^ (1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(1/2 - 1i/2) - 2^(1/2)* log(2^(1/2)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(4 + 4i ) - exp(x)/2)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(1/2 + 1i/2)
Time = 0.28 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.73 \[ \int e^x \text {sech}^2(4 x) \, dx=\frac {-2 e^{8 x} \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )-2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )+2 e^{8 x} \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )+2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )-2 e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )-2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )+2 e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )+2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )-e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )+e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )-\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )+\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )-e^{8 x} \sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )+e^{8 x} \sqrt {\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )-\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )+\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )-16 e^{x}}{32 e^{8 x}+32} \] Input:
int(exp(x)*sech(4*x)^2,x)
Output:
( - 2*e**(8*x)*sqrt(sqrt(2) + 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*e**(8*x)*sqrt(sqrt(2) + 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*e**(8*x)*sqrt( - sqrt(2) + 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*e**(8*x)*sq rt( - sqrt(2) + 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)) - e**(8*x)*sqrt( - sqrt(2) + 2)*log( - e**x*sqrt( - sqrt(2) + 2) + e**(2*x) + 1) + e**(8*x)*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)*log(e**x*sqrt( - sqrt(2) + 2) + e* *(2*x) + 1) - e**(8*x)*sqrt(sqrt(2) + 2)*log( - e**x*sqrt(sqrt(2) + 2) + e **(2*x) + 1) + e**(8*x)*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)*log(e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) - 16*e** x)/(32*(e**(8*x) + 1))