Integrand size = 22, antiderivative size = 77 \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=-\frac {2 e^{a-d}}{b \left (1+e^{2 d+2 b x}\right )^2}+\frac {4 e^{a-d}}{b \left (1+e^{2 d+2 b x}\right )}+\frac {e^{a-d} \log \left (1+e^{2 d+2 b x}\right )}{b} \] Output:
-2*exp(a-d)/b/(1+exp(2*b*x+2*d))^2+4*exp(a-d)/b/(1+exp(2*b*x+2*d))+exp(a-d )*ln(1+exp(2*b*x+2*d))/b
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73 \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=\frac {2 e^{a-d} \left (\frac {1+2 e^{2 (d+b x)}}{\left (1+e^{2 (d+b x)}\right )^2}+\frac {1}{2} \log \left (1+e^{2 (d+b x)}\right )\right )}{b} \] Input:
Integrate[E^(a + b*x)*Sech[d + b*x]*Tanh[d + b*x]^2,x]
Output:
(2*E^(a - d)*((1 + 2*E^(2*(d + b*x)))/(1 + E^(2*(d + b*x)))^2 + Log[1 + E^ (2*(d + b*x))]/2))/b
Time = 0.38 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2720, 27, 353, 49, 2009}
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.
\(\displaystyle \int e^{a+b x} \tanh ^2(b x+d) \text {sech}(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {2 e^{a+b x} \left (1-e^{2 b x}\right )^2}{\left (1+e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 e^a \int \frac {e^{b x} \left (1-e^{2 b x}\right )^2}{\left (1+e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {e^a \int \frac {\left (1-e^{2 b x}\right )^2}{\left (1+e^{2 b x}\right )^3}de^{2 b x}}{b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {e^a \int \left (\frac {1}{1+e^{2 b x}}-\frac {4}{\left (1+e^{2 b x}\right )^2}+\frac {4}{\left (1+e^{2 b x}\right )^3}\right )de^{2 b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^a \left (\frac {4}{e^{2 b x}+1}-\frac {2}{\left (e^{2 b x}+1\right )^2}+\log \left (e^{2 b x}+1\right )\right )}{b}\) |
Input:
Int[E^(a + b*x)*Sech[d + b*x]*Tanh[d + b*x]^2,x]
Output:
(E^a*(-2/(1 + E^(2*b*x))^2 + 4/(1 + E^(2*b*x)) + Log[1 + E^(2*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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 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]]]
Time = 1.96 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{a -d} a}{b}+\frac {2 \left (2 \,{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) {\mathrm e}^{3 a -d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{a -d}}{b}\) | \(94\) |
Input:
int(exp(b*x+a)*sech(b*x+d)*tanh(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-2/b*exp(a-d)*a+2/(exp(2*b*x+2*a+2*d)+exp(2*a))^2/b*(2*exp(2*b*x+2*a+2*d)+ exp(2*a))*exp(3*a-d)+ln(exp(2*b*x+2*a)+exp(2*a-2*d))/b*exp(a-d)
Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (71) = 142\).
Time = 0.09 (sec) , antiderivative size = 477, normalized size of antiderivative = 6.19 \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=\frac {4 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + 4 \, {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} + {\left (\cosh \left (b x + d\right )^{4} \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{4} + 4 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{3} + 2 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + 2 \, {\left (3 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) - {\left (3 \, \cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-a + d\right ) + \cosh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} + 4 \, {\left (\cosh \left (b x + d\right )^{3} \cosh \left (-a + d\right ) + \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - {\left (\cosh \left (b x + d\right )^{3} + \cosh \left (b x + d\right )\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{4} + 2 \, \cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-a + d\right ) + \cosh \left (-a + d\right )\right )} \log \left (\frac {2 \, \cosh \left (b x + d\right )}{\cosh \left (b x + d\right ) - \sinh \left (b x + d\right )}\right ) + 8 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - 2 \, {\left (2 \, \cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-a + d\right ) + 2 \, \cosh \left (-a + d\right )}{b \cosh \left (b x + d\right )^{4} + 4 \, b \cosh \left (b x + d\right ) \sinh \left (b x + d\right )^{3} + b \sinh \left (b x + d\right )^{4} + 2 \, b \cosh \left (b x + d\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + d\right )^{2} + b\right )} \sinh \left (b x + d\right )^{2} + 4 \, {\left (b \cosh \left (b x + d\right )^{3} + b \cosh \left (b x + d\right )\right )} \sinh \left (b x + d\right ) + b} \] Input:
integrate(exp(b*x+a)*sech(b*x+d)*tanh(b*x+d)^2,x, algorithm="fricas")
Output:
(4*cosh(b*x + d)^2*cosh(-a + d) + 4*(cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^2 + (cosh(b*x + d)^4*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*si nh(b*x + d)^4 + 4*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d) )*sinh(b*x + d)^3 + 2*cosh(b*x + d)^2*cosh(-a + d) + 2*(3*cosh(b*x + d)^2* cosh(-a + d) - (3*cosh(b*x + d)^2 + 1)*sinh(-a + d) + cosh(-a + d))*sinh(b *x + d)^2 + 4*(cosh(b*x + d)^3*cosh(-a + d) + cosh(b*x + d)*cosh(-a + d) - (cosh(b*x + d)^3 + cosh(b*x + d))*sinh(-a + d))*sinh(b*x + d) - (cosh(b*x + d)^4 + 2*cosh(b*x + d)^2 + 1)*sinh(-a + d) + cosh(-a + d))*log(2*cosh(b *x + d)/(cosh(b*x + d) - sinh(b*x + d))) + 8*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d) - 2*(2*cosh(b*x + d)^2 + 1)*sin h(-a + d) + 2*cosh(-a + d))/(b*cosh(b*x + d)^4 + 4*b*cosh(b*x + d)*sinh(b* x + d)^3 + b*sinh(b*x + d)^4 + 2*b*cosh(b*x + d)^2 + 2*(3*b*cosh(b*x + d)^ 2 + b)*sinh(b*x + d)^2 + 4*(b*cosh(b*x + d)^3 + b*cosh(b*x + d))*sinh(b*x + d) + b)
\[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=e^{a} \int e^{b x} \tanh ^{2}{\left (b x + d \right )} \operatorname {sech}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*sech(b*x+d)*tanh(b*x+d)**2,x)
Output:
exp(a)*Integral(exp(b*x)*tanh(b*x + d)**2*sech(b*x + d), x)
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=\frac {e^{\left (a - d\right )} \log \left (e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + e^{\left (2 \, a\right )}\right )}{b} + \frac {2 \, {\left (2 \, e^{\left (2 \, b x + 5 \, a + 2 \, d\right )} + e^{\left (5 \, a\right )}\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a + 5 \, d\right )} + 2 \, e^{\left (2 \, b x + 4 \, a + 3 \, d\right )} + e^{\left (4 \, a + d\right )}\right )}} \] Input:
integrate(exp(b*x+a)*sech(b*x+d)*tanh(b*x+d)^2,x, algorithm="maxima")
Output:
e^(a - d)*log(e^(2*b*x + 2*a + 2*d) + e^(2*a))/b + 2*(2*e^(2*b*x + 5*a + 2 *d) + e^(5*a))/(b*(e^(4*b*x + 4*a + 5*d) + 2*e^(2*b*x + 4*a + 3*d) + e^(4* a + d)))
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.19 \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=\frac {2 \, e^{\left (a - d\right )} \log \left (e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + e^{\left (2 \, a\right )}\right ) - \frac {{\left (3 \, e^{\left (4 \, b x + 5 \, a + 4 \, d\right )} - 2 \, e^{\left (2 \, b x + 5 \, a + 2 \, d\right )} - e^{\left (5 \, a\right )}\right )} e^{\left (-d\right )}}{{\left (e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + e^{\left (2 \, a\right )}\right )}^{2}}}{2 \, b} \] Input:
integrate(exp(b*x+a)*sech(b*x+d)*tanh(b*x+d)^2,x, algorithm="giac")
Output:
1/2*(2*e^(a - d)*log(e^(2*b*x + 2*a + 2*d) + e^(2*a)) - (3*e^(4*b*x + 5*a + 4*d) - 2*e^(2*b*x + 5*a + 2*d) - e^(5*a))*e^(-d)/(e^(2*b*x + 2*a + 2*d) + e^(2*a))^2)/b
Timed out. \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}\,{\mathrm {tanh}\left (d+b\,x\right )}^2}{\mathrm {cosh}\left (d+b\,x\right )} \,d x \] Input:
int((exp(a + b*x)*tanh(d + b*x)^2)/cosh(d + b*x),x)
Output:
int((exp(a + b*x)*tanh(d + b*x)^2)/cosh(d + b*x), x)
Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int e^{a+b x} \text {sech}(d+b x) \tanh ^2(d+b x) \, dx=\frac {e^{a} \left (e^{4 b x +4 d} \mathrm {log}\left (e^{2 b x +2 d}+1\right )-2 e^{4 b x +4 d}+2 e^{2 b x +2 d} \mathrm {log}\left (e^{2 b x +2 d}+1\right )+\mathrm {log}\left (e^{2 b x +2 d}+1\right )\right )}{e^{d} b \left (e^{4 b x +4 d}+2 e^{2 b x +2 d}+1\right )} \] Input:
int(exp(b*x+a)*sech(b*x+d)*tanh(b*x+d)^2,x)
Output:
(e**a*(e**(4*b*x + 4*d)*log(e**(2*b*x + 2*d) + 1) - 2*e**(4*b*x + 4*d) + 2 *e**(2*b*x + 2*d)*log(e**(2*b*x + 2*d) + 1) + log(e**(2*b*x + 2*d) + 1)))/ (e**d*b*(e**(4*b*x + 4*d) + 2*e**(2*b*x + 2*d) + 1))