Integrand size = 22, antiderivative size = 71 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=\frac {2 e^{2 a-d+b x}}{b}+\frac {2 e^{2 a-d+b x}}{b \left (1+e^{2 d+2 b x}\right )}-\frac {4 e^{2 a-2 d} \arctan \left (e^{d+b x}\right )}{b} \] Output:
2*exp(b*x+2*a-d)/b+2*exp(b*x+2*a-d)/b/(1+exp(2*b*x+2*d))-4*exp(2*a-2*d)*ar ctan(exp(b*x+d))/b
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.46 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=\frac {2 e^{2 a} \left (e^{b x} \cosh (d)-2 \arctan \left (e^{b x} (\cosh (d)+\sinh (d))\right ) \cosh (2 d)-e^{b x} \sinh (d)+\frac {e^{b x} (\cosh (d)-\sinh (d))^2}{\left (1+e^{2 b x}\right ) \cosh (d)+\left (-1+e^{2 b x}\right ) \sinh (d)}+2 \arctan \left (e^{b x} (\cosh (d)+\sinh (d))\right ) \sinh (2 d)\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))*Sech[d + b*x]*Tanh[d + b*x],x]
Output:
(2*E^(2*a)*(E^(b*x)*Cosh[d] - 2*ArcTan[E^(b*x)*(Cosh[d] + Sinh[d])]*Cosh[2 *d] - E^(b*x)*Sinh[d] + (E^(b*x)*(Cosh[d] - Sinh[d])^2)/((1 + E^(2*b*x))*C osh[d] + (-1 + E^(2*b*x))*Sinh[d]) + 2*ArcTan[E^(b*x)*(Cosh[d] + Sinh[d])] *Sinh[2*d]))/b
Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2720, 27, 360, 27, 299, 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.
\(\displaystyle \int e^{2 (a+b x)} \tanh (b x+d) \text {sech}(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {2 e^{2 a+2 b x} \left (1-e^{2 b x}\right )}{\left (1+e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 e^{2 a} \int \frac {e^{2 b x} \left (1-e^{2 b x}\right )}{\left (1+e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 360 |
\(\displaystyle -\frac {2 e^{2 a} \left (-\frac {1}{2} \int -\frac {2 \left (1-e^{2 b x}\right )}{1+e^{2 b x}}de^{b x}-\frac {e^{b x}}{e^{2 b x}+1}\right )}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 e^{2 a} \left (\int \frac {1-e^{2 b x}}{1+e^{2 b x}}de^{b x}-\frac {e^{b x}}{e^{2 b x}+1}\right )}{b}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {2 e^{2 a} \left (2 \int \frac {1}{1+e^{2 b x}}de^{b x}-e^{b x}-\frac {e^{b x}}{e^{2 b x}+1}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {2 e^{2 a} \left (2 \arctan \left (e^{b x}\right )-e^{b x}-\frac {e^{b x}}{e^{2 b x}+1}\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))*Sech[d + b*x]*Tanh[d + b*x],x]
Output:
(-2*E^(2*a)*(-E^(b*x) - E^(b*x)/(1 + E^(2*b*x)) + 2*ArcTan[E^(b*x)]))/b
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[(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)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 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 complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.62
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +2 a -d}}{b}+\frac {2 \,{\mathrm e}^{b x +4 a -d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) b}+\frac {2 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {2 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}\) | \(115\) |
Input:
int(exp(2*b*x+2*a)*sech(b*x+d)*tanh(b*x+d),x,method=_RETURNVERBOSE)
Output:
2*exp(b*x+2*a-d)/b+2/(exp(2*b*x+2*a+2*d)+exp(2*a))/b*exp(b*x+4*a-d)+2*I*ln (exp(b*x+a)-I*exp(a-d))/b*exp(2*a-2*d)-2*I*ln(exp(b*x+a)+I*exp(a-d))/b*exp (2*a-2*d)
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (66) = 132\).
Time = 0.09 (sec) , antiderivative size = 360, normalized size of antiderivative = 5.07 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=\frac {2 \, {\left (\cosh \left (b x + d\right )^{3} \cosh \left (-2 \, a + 2 \, d\right ) + {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{3} + 3 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - \cosh \left (b x + d\right ) \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{2} - 2 \, {\left (\cosh \left (b x + d\right )^{2} \cosh \left (-2 \, a + 2 \, d\right ) + {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{2} + 2 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - \cosh \left (b x + d\right ) \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-2 \, a + 2 \, d\right ) + \cosh \left (-2 \, a + 2 \, d\right )\right )} \arctan \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right )\right ) + 2 \, \cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) + {\left (3 \, \cosh \left (b x + d\right )^{2} \cosh \left (-2 \, a + 2 \, d\right ) - {\left (3 \, \cosh \left (b x + d\right )^{2} + 2\right )} \sinh \left (-2 \, a + 2 \, d\right ) + 2 \, \cosh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{3} + 2 \, \cosh \left (b x + d\right )\right )} \sinh \left (-2 \, a + 2 \, d\right )\right )}}{b \cosh \left (b x + d\right )^{2} + 2 \, b \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + b \sinh \left (b x + d\right )^{2} + b} \] Input:
integrate(exp(2*b*x+2*a)*sech(b*x+d)*tanh(b*x+d),x, algorithm="fricas")
Output:
2*(cosh(b*x + d)^3*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d) )*sinh(b*x + d)^3 + 3*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh (-2*a + 2*d))*sinh(b*x + d)^2 - 2*(cosh(b*x + d)^2*cosh(-2*a + 2*d) + (cos h(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d)^2 + 2*(cosh(b*x + d)*cosh( -2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d) - (cosh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + cosh(-2*a + 2*d))*arctan(cosh(b*x + d) + sinh (b*x + d)) + 2*cosh(b*x + d)*cosh(-2*a + 2*d) + (3*cosh(b*x + d)^2*cosh(-2 *a + 2*d) - (3*cosh(b*x + d)^2 + 2)*sinh(-2*a + 2*d) + 2*cosh(-2*a + 2*d)) *sinh(b*x + d) - (cosh(b*x + d)^3 + 2*cosh(b*x + d))*sinh(-2*a + 2*d))/(b* cosh(b*x + d)^2 + 2*b*cosh(b*x + d)*sinh(b*x + d) + b*sinh(b*x + d)^2 + b)
\[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=e^{2 a} \int e^{2 b x} \tanh {\left (b x + d \right )} \operatorname {sech}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*b*x+2*a)*sech(b*x+d)*tanh(b*x+d),x)
Output:
exp(2*a)*Integral(exp(2*b*x)*tanh(b*x + d)*sech(b*x + d), x)
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=\frac {4 \, \arctan \left (e^{\left (-b x - d\right )}\right ) e^{\left (2 \, a - 2 \, d\right )}}{b} + \frac {2 \, {\left (2 \, e^{\left (-2 \, b x - 2 \, d\right )} + 1\right )} e^{\left (2 \, a - 2 \, d\right )}}{b {\left (e^{\left (-b x - d\right )} + e^{\left (-3 \, b x - 3 \, d\right )}\right )}} \] Input:
integrate(exp(2*b*x+2*a)*sech(b*x+d)*tanh(b*x+d),x, algorithm="maxima")
Output:
4*arctan(e^(-b*x - d))*e^(2*a - 2*d)/b + 2*(2*e^(-2*b*x - 2*d) + 1)*e^(2*a - 2*d)/(b*(e^(-b*x - d) + e^(-3*b*x - 3*d)))
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=-\frac {2 \, {\left (2 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (2 \, a - 2 \, d\right )} - \frac {e^{\left (b x + 2 \, a - d\right )}}{e^{\left (2 \, b x + 2 \, d\right )} + 1} - e^{\left (b x + 2 \, a - d\right )}\right )}}{b} \] Input:
integrate(exp(2*b*x+2*a)*sech(b*x+d)*tanh(b*x+d),x, algorithm="giac")
Output:
-2*(2*arctan(e^(b*x + d))*e^(2*a - 2*d) - e^(b*x + 2*a - d)/(e^(2*b*x + 2* d) + 1) - e^(b*x + 2*a - d))/b
Time = 2.71 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.35 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b}+\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b\,\left ({\mathrm {e}}^{2\,d+2\,b\,x}+1\right )}-\frac {4\,\sqrt {{\mathrm {e}}^{4\,a-4\,d}}\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-d}\,{\mathrm {e}}^{b\,x}\,\sqrt {b^2}}{b\,\sqrt {{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}}}\right )}{\sqrt {b^2}} \] Input:
int((exp(2*a + 2*b*x)*tanh(d + b*x))/cosh(d + b*x),x)
Output:
(2*exp(2*a - d + b*x))/b + (2*exp(2*a - d + b*x))/(b*(exp(2*d + 2*b*x) + 1 )) - (4*exp(4*a - 4*d)^(1/2)*atan((exp(2*a)*exp(-d)*exp(b*x)*(b^2)^(1/2))/ (b*(exp(4*a)*exp(-4*d))^(1/2))))/(b^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14 \[ \int e^{2 (a+b x)} \text {sech}(d+b x) \tanh (d+b x) \, dx=\frac {2 e^{2 a} \left (-2 e^{2 b x +2 d} \mathit {atan} \left (e^{b x +d}\right )-2 \mathit {atan} \left (e^{b x +d}\right )+e^{3 b x +3 d}+2 e^{b x +d}\right )}{e^{2 d} b \left (e^{2 b x +2 d}+1\right )} \] Input:
int(exp(2*b*x+2*a)*sech(b*x+d)*tanh(b*x+d),x)
Output:
(2*e**(2*a)*( - 2*e**(2*b*x + 2*d)*atan(e**(b*x + d)) - 2*atan(e**(b*x + d )) + e**(3*b*x + 3*d) + 2*e**(b*x + d)))/(e**(2*d)*b*(e**(2*b*x + 2*d) + 1 ))