Integrand size = 8, antiderivative size = 55 \[ \int e^x \text {sech}(3 x) \, dx=-\frac {\arctan \left (\frac {1-2 e^{2 x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1+e^{2 x}\right )+\frac {1}{6} \log \left (1-e^{2 x}+e^{4 x}\right ) \] Output:
-1/3*arctan(1/3*(1-2*exp(2*x))*3^(1/2))*3^(1/2)-1/3*ln(1+exp(2*x))+1/6*ln( 1-exp(2*x)+exp(4*x))
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 = 0.44 \[ \int e^x \text {sech}(3 x) \, dx=\frac {1}{2} e^{4 x} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-e^{6 x}\right ) \] Input:
Integrate[E^x*Sech[3*x],x]
Output:
(E^(4*x)*Hypergeometric2F1[2/3, 1, 5/3, -E^(6*x)])/2
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {2720, 27, 807, 821, 16, 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.
\(\displaystyle \int e^x \text {sech}(3 x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {2 e^{3 x}}{e^{6 x}+1}de^x\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {e^{3 x}}{1+e^{6 x}}de^x\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \int \frac {e^{2 x}}{e^{3 x}+1}de^{2 x}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {1}{3} \int \left (1+e^{2 x}\right )de^{2 x}-\frac {1}{3} \int \frac {1}{1+e^{2 x}}de^{2 x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{3} \int \left (1+e^{2 x}\right )de^{2 x}-\frac {1}{3} \log \left (e^{2 x}+1\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int 1de^{2 x}}{2}+\frac {1}{2} \int \left (-1+2 e^{2 x}\right )de^{2 x}\right )-\frac {1}{3} \log \left (e^{2 x}+1\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \int 1de^{2 x}}{2}-\frac {1}{2} \int \left (1-2 e^{2 x}\right )de^{2 x}\right )-\frac {1}{3} \log \left (e^{2 x}+1\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{3} \left (-3 \int \frac {1}{-2-2 e^{2 x}}d\left (-1+2 e^{2 x}\right )-\frac {1}{2} \int \left (1-2 e^{2 x}\right )de^{2 x}\right )-\frac {1}{3} \log \left (e^{2 x}+1\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 e^{2 x}-1}{\sqrt {3}}\right )-\frac {1}{2} \int \left (1-2 e^{2 x}\right )de^{2 x}\right )-\frac {1}{3} \log \left (e^{2 x}+1\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\arctan \left (\frac {2 e^{2 x}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (e^{2 x}+1\right )\) |
Input:
Int[E^x*Sech[3*x],x]
Output:
ArcTan[(-1 + 2*E^(2*x))/Sqrt[3]]/Sqrt[3] - Log[1 + E^(2*x)]/3
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, 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[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 complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.44
method | result | size |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 x}+1\right )}{3}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}+\frac {i \ln \left ({\mathrm e}^{2 x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}-\frac {i \ln \left ({\mathrm e}^{2 x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{6}\) | \(79\) |
Input:
int(exp(x)*sech(3*x),x,method=_RETURNVERBOSE)
Output:
-1/3*ln(exp(2*x)+1)+1/6*ln(exp(2*x)-1/2+1/2*I*3^(1/2))+1/6*I*ln(exp(2*x)-1 /2+1/2*I*3^(1/2))*3^(1/2)+1/6*ln(exp(2*x)-1/2-1/2*I*3^(1/2))-1/6*I*ln(exp( 2*x)-1/2-1/2*I*3^(1/2))*3^(1/2)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.51 \[ \int e^x \text {sech}(3 x) \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} \cosh \left (x\right ) + 3 \, \sqrt {3} \sinh \left (x\right )}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) + \frac {1}{6} \, \log \left (\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \] Input:
integrate(exp(x)*sech(3*x),x, algorithm="fricas")
Output:
-1/3*sqrt(3)*arctan(-1/3*(sqrt(3)*cosh(x) + 3*sqrt(3)*sinh(x))/(cosh(x) - sinh(x))) + 1/6*log((2*cosh(x)^2 + 2*sinh(x)^2 - 1)/(cosh(x)^2 - 2*cosh(x) *sinh(x) + sinh(x)^2)) - 1/3*log(2*cosh(x)/(cosh(x) - sinh(x)))
\[ \int e^x \text {sech}(3 x) \, dx=\int e^{x} \operatorname {sech}{\left (3 x \right )}\, dx \] Input:
integrate(exp(x)*sech(3*x),x)
Output:
Integral(exp(x)*sech(3*x), x)
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.29 \[ \int e^x \text {sech}(3 x) \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\sqrt {3} + 2 \, e^{x}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\sqrt {3} + 2 \, e^{x}\right ) + \frac {1}{6} \, \log \left (\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{6} \, \log \left (-\sqrt {3} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \] Input:
integrate(exp(x)*sech(3*x),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(sqrt(3) + 2*e^x) + 1/3*sqrt(3)*arctan(-sqrt(3) + 2*e^x ) + 1/6*log(sqrt(3)*e^x + e^(2*x) + 1) + 1/6*log(-sqrt(3)*e^x + e^(2*x) + 1) - 1/3*log(e^(2*x) + 1)
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int e^x \text {sech}(3 x) \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (2 \, x\right )} - 1\right )}\right ) + \frac {1}{6} \, \log \left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{3} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \] Input:
integrate(exp(x)*sech(3*x),x, algorithm="giac")
Output:
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(2*x) - 1)) + 1/6*log(e^(4*x) - e^(2*x ) + 1) - 1/3*log(e^(2*x) + 1)
Time = 2.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int e^x \text {sech}(3 x) \, dx=-\frac {\ln \left (8\,{\mathrm {e}}^{2\,x}+8\right )}{3}-\ln \left (24\,{\mathrm {e}}^{2\,x}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+8\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (8-24\,{\mathrm {e}}^{2\,x}\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \] Input:
int(exp(x)/cosh(3*x),x)
Output:
log(8 - 24*exp(2*x)*((3^(1/2)*1i)/6 + 1/6))*((3^(1/2)*1i)/6 + 1/6) - log(2 4*exp(2*x)*((3^(1/2)*1i)/6 - 1/6) + 8)*((3^(1/2)*1i)/6 - 1/6) - log(8*exp( 2*x) + 8)/3
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.31 \[ \int e^x \text {sech}(3 x) \, dx=\frac {\sqrt {3}\, \mathit {atan} \left (2 e^{x}-\sqrt {3}\right )}{3}-\frac {\sqrt {3}\, \mathit {atan} \left (2 e^{x}+\sqrt {3}\right )}{3}+\frac {\mathrm {log}\left (e^{2 x}-e^{x} \sqrt {3}+1\right )}{6}+\frac {\mathrm {log}\left (e^{2 x}+e^{x} \sqrt {3}+1\right )}{6}-\frac {\mathrm {log}\left (e^{2 x}+1\right )}{3} \] Input:
int(exp(x)*sech(3*x),x)
Output:
(2*sqrt(3)*atan(2*e**x - sqrt(3)) - 2*sqrt(3)*atan(2*e**x + sqrt(3)) + log (e**(2*x) - e**x*sqrt(3) + 1) + log(e**(2*x) + e**x*sqrt(3) + 1) - 2*log(e **(2*x) + 1))/6