Integrand size = 22, antiderivative size = 50 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {2 e^{a-d}}{b \left (1+e^{2 d+2 b x}\right )}-\frac {2 e^{a-d} \text {arctanh}\left (e^{2 d+2 b x}\right )}{b} \] Output:
-2*exp(a-d)/b/(1+exp(2*b*x+2*d))-2*exp(a-d)*arctanh(exp(2*b*x+2*d))/b
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\frac {8 e^{a-d} \left (-\frac {1}{4 \left (1+e^{2 (d+b x)}\right )}-\frac {1}{4} \text {arctanh}\left (e^{2 (d+b x)}\right )\right )}{b} \] Input:
Integrate[E^(a + b*x)*Csch[d + b*x]*Sech[d + b*x]^2,x]
Output:
(8*E^(a - d)*(-1/4*1/(1 + E^(2*(d + b*x))) - ArcTanh[E^(2*(d + b*x))]/4))/ b
Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2720, 27, 354, 86, 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.
\(\displaystyle \int e^{a+b x} \text {csch}(b x+d) \text {sech}^2(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {8 e^{a+3 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {8 e^a \int \frac {e^{3 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {4 e^a \int \frac {e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{2 b x}}{b}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {4 e^a \int \left (-\frac {1}{2 \left (1+e^{2 b x}\right )^2}-\frac {1}{2 \left (-1+e^{2 b x}\right )}\right )de^{2 b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 e^a \left (\frac {1}{2} \text {arctanh}\left (e^{2 b x}\right )+\frac {1}{2 \left (e^{2 b x}+1\right )}\right )}{b}\) |
Input:
Int[E^(a + b*x)*Csch[d + b*x]*Sech[d + b*x]^2,x]
Output:
(-4*E^a*(1/(2*(1 + E^(2*b*x))) + ArcTanh[E^(2*b*x)]/2))/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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && IntegerQ [(m - 1)/2]
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]]]
Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(46)=92\).
Time = 13.78 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.90
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{3 a -d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{a -d}}{b}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{a -d}}{b}\) | \(95\) |
Input:
int(exp(b*x+a)*csch(b*x+d)*sech(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-2/(exp(2*b*x+2*a+2*d)+exp(2*a))/b*exp(3*a-d)+ln(exp(2*b*x+2*a)-exp(2*a-2* d))/b*exp(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. 313 vs. \(2 (46) = 92\).
Time = 0.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 6.26 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {{\left (\cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} + 2 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-a + d\right ) + \cosh \left (-a + d\right )\right )} \log \left (\frac {2 \, \cosh \left (b x + d\right )}{\cosh \left (b x + d\right ) - \sinh \left (b x + d\right )}\right ) - {\left (\cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} + 2 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-a + d\right ) + \cosh \left (-a + d\right )\right )} \log \left (\frac {2 \, \sinh \left (b x + d\right )}{\cosh \left (b x + d\right ) - \sinh \left (b x + d\right )}\right ) + 2 \, \cosh \left (-a + d\right ) - 2 \, \sinh \left (-a + d\right )}{b \cosh \left (b x + d\right )^{2} + 2 \, b \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + b \sinh \left (b x + d\right )^{2} + b} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d)^2,x, algorithm="fricas")
Output:
-((cosh(b*x + d)^2*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^2 + 2*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b *x + d) - (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))) - (cosh(b*x + d)^2*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^2 + 2*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d) - (cosh(b*x + d)^2 + 1)*si nh(-a + d) + cosh(-a + d))*log(2*sinh(b*x + d)/(cosh(b*x + d) - sinh(b*x + d))) + 2*cosh(-a + d) - 2*sinh(-a + d))/(b*cosh(b*x + d)^2 + 2*b*cosh(b*x + d)*sinh(b*x + d) + b*sinh(b*x + d)^2 + b)
\[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}{\left (b x + d \right )} \operatorname {sech}^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d)**2,x)
Output:
exp(a)*Integral(exp(b*x)*csch(b*x + d)*sech(b*x + d)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (46) = 92\).
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.08 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {e^{\left (a - d\right )} \log \left (e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + e^{\left (2 \, a\right )}\right )}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{b} - \frac {2 \, e^{\left (3 \, a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a + 3 \, d\right )} + e^{\left (2 \, a + d\right )}\right )}} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d)^2,x, algorithm="maxima")
Output:
-e^(a - d)*log(e^(2*b*x + 2*a + 2*d) + e^(2*a))/b + e^(a - d)*log(e^(b*x + a + d) + e^a)/b + e^(a - d)*log(e^(b*x + a + d) - e^a)/b - 2*e^(3*a)/(b*( e^(2*b*x + 2*a + 3*d) + e^(2*a + d)))
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.46 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=-{\left (\frac {e^{\left (-4 \, d\right )} \log \left (e^{\left (2 \, b x + 2 \, d\right )} + 1\right )}{b} - \frac {e^{\left (-4 \, d\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, d\right )} - 1 \right |}\right )}{b} + \frac {2 \, e^{\left (-4 \, d\right )}}{b {\left (e^{\left (2 \, b x + 2 \, d\right )} + 1\right )}}\right )} e^{\left (a + 3 \, d\right )} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d)^2,x, algorithm="giac")
Output:
-(e^(-4*d)*log(e^(2*b*x + 2*d) + 1)/b - e^(-4*d)*log(abs(e^(2*b*x + 2*d) - 1))/b + 2*e^(-4*d)/(b*(e^(2*b*x + 2*d) + 1)))*e^(a + 3*d)
Timed out. \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (d+b\,x\right )}^2\,\mathrm {sinh}\left (d+b\,x\right )} \,d x \] Input:
int(exp(a + b*x)/(cosh(d + b*x)^2*sinh(d + b*x)),x)
Output:
int(exp(a + b*x)/(cosh(d + b*x)^2*sinh(d + b*x)), x)
Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.82 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\frac {e^{a} \left (e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )+e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )-e^{2 b x +2 d} \mathrm {log}\left (e^{2 b x +2 d}+1\right )+2 e^{2 b x +2 d}+\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )-\mathrm {log}\left (e^{2 b x +2 d}+1\right )\right )}{e^{d} b \left (e^{2 b x +2 d}+1\right )} \] Input:
int(exp(b*x+a)*csch(b*x+d)*sech(b*x+d)^2,x)
Output:
(e**a*(e**(2*b*x + 2*d)*log(e**(b*x + d) - 1) + e**(2*b*x + 2*d)*log(e**(b *x + d) + 1) - e**(2*b*x + 2*d)*log(e**(2*b*x + 2*d) + 1) + 2*e**(2*b*x + 2*d) + log(e**(b*x + d) - 1) + log(e**(b*x + d) + 1) - log(e**(2*b*x + 2*d ) + 1)))/(e**d*b*(e**(2*b*x + 2*d) + 1))