Integrand size = 24, antiderivative size = 82 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=\frac {2 e^{2 a-2 d}}{b \left (1+e^{2 d+2 b x}\right )^2}-\frac {6 e^{2 a-2 d}}{b \left (1+e^{2 d+2 b x}\right )}-\frac {2 e^{2 a-2 d} \text {arctanh}\left (e^{2 d+2 b x}\right )}{b} \] Output:
2*exp(2*a-2*d)/b/(1+exp(2*b*x+2*d))^2-6*exp(2*a-2*d)/b/(1+exp(2*b*x+2*d))- 2*exp(2*a-2*d)*arctanh(exp(2*b*x+2*d))/b
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=\frac {16 e^{2 a-2 d} \left (\frac {1}{8 \left (1+e^{2 (d+b x)}\right )^2}-\frac {3}{8 \left (1+e^{2 (d+b x)}\right )}-\frac {1}{8} \text {arctanh}\left (e^{2 (d+b x)}\right )\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))*Csch[d + b*x]*Sech[d + b*x]^3,x]
Output:
(16*E^(2*a - 2*d)*(1/(8*(1 + E^(2*(d + b*x)))^2) - 3/(8*(1 + E^(2*(d + b*x )))) - ArcTanh[E^(2*(d + b*x))]/8))/b
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.61, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2720, 27, 354, 99, 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.
\(\displaystyle \int e^{2 (a+b x)} \text {csch}(b x+d) \text {sech}^3(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {16 e^{2 a+5 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {16 e^{2 a} \int \frac {e^{5 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {8 e^{2 a} \int \frac {e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^3}de^{2 b x}}{b}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {8 e^{2 a} \int \left (-\frac {3}{4 \left (1+e^{2 b x}\right )^2}+\frac {1}{2 \left (1+e^{2 b x}\right )^3}-\frac {1}{4 \left (-1+e^{2 b x}\right )}\right )de^{2 b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8 e^{2 a} \left (\frac {1}{4} \text {arctanh}\left (e^{2 b x}\right )+\frac {3}{4 \left (e^{2 b x}+1\right )}-\frac {1}{4 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))*Csch[d + b*x]*Sech[d + b*x]^3,x]
Output:
(-8*E^(2*a)*(-1/4*1/(1 + E^(2*b*x))^2 + 3/(4*(1 + E^(2*b*x))) + ArcTanh[E^ (2*b*x)]/4))/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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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]]]
Time = 106.35 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {2 \left (3 \,{\mathrm e}^{2 b x +2 a +2 d}+2 \,{\mathrm e}^{2 a}\right ) {\mathrm e}^{4 a -2 d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{2 a -2 d}}{b}\) | \(120\) |
Input:
int(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-2/(exp(2*b*x+2*a+2*d)+exp(2*a))^2/b*(3*exp(2*b*x+2*a+2*d)+2*exp(2*a))*exp (4*a-2*d)+ln(exp(2*b*x+2*a)-exp(2*a-2*d))/b*exp(2*a-2*d)-ln(exp(2*b*x+2*a) +exp(2*a-2*d))/b*exp(2*a-2*d)
Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (76) = 152\).
Time = 0.09 (sec) , antiderivative size = 817, normalized size of antiderivative = 9.96 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^3,x, algorithm="fricas")
Output:
-(6*cosh(b*x + d)^2*cosh(-2*a + 2*d) + 6*(cosh(-2*a + 2*d) - sinh(-2*a + 2 *d))*sinh(b*x + d)^2 + (cosh(b*x + d)^4*cosh(-2*a + 2*d) + (cosh(-2*a + 2* d) - sinh(-2*a + 2*d))*sinh(b*x + d)^4 + 4*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d)^3 + 2*cosh(b*x + d)^2*cos h(-2*a + 2*d) + 2*(3*cosh(b*x + d)^2*cosh(-2*a + 2*d) - (3*cosh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + cosh(-2*a + 2*d))*sinh(b*x + d)^2 + 4*(cosh(b*x + d)^3*cosh(-2*a + 2*d) + cosh(b*x + d)*cosh(-2*a + 2*d) - (cosh(b*x + d)^3 + cosh(b*x + d))*sinh(-2*a + 2*d))*sinh(b*x + d) - (cosh(b*x + d)^4 + 2*c osh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + cosh(-2*a + 2*d))*log(2*cosh(b*x + d)/(cosh(b*x + d) - sinh(b*x + d))) - (cosh(b*x + d)^4*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d)^4 + 4*(cosh(b*x + d)*c osh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d)^3 + 2*cosh (b*x + d)^2*cosh(-2*a + 2*d) + 2*(3*cosh(b*x + d)^2*cosh(-2*a + 2*d) - (3* cosh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + cosh(-2*a + 2*d))*sinh(b*x + d)^2 + 4*(cosh(b*x + d)^3*cosh(-2*a + 2*d) + cosh(b*x + d)*cosh(-2*a + 2*d) - ( cosh(b*x + d)^3 + cosh(b*x + d))*sinh(-2*a + 2*d))*sinh(b*x + d) - (cosh(b *x + d)^4 + 2*cosh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + cosh(-2*a + 2*d))*lo g(2*sinh(b*x + d)/(cosh(b*x + d) - sinh(b*x + d))) + 12*(cosh(b*x + d)*cos h(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d) - 2*(3*cosh( b*x + d)^2 + 2)*sinh(-2*a + 2*d) + 4*cosh(-2*a + 2*d))/(b*cosh(b*x + d)...
\[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=e^{2 a} \int e^{2 b x} \operatorname {csch}{\left (b x + d \right )} \operatorname {sech}^{3}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)**3,x)
Output:
exp(2*a)*Integral(exp(2*b*x)*csch(b*x + d)*sech(b*x + d)**3, x)
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.49 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=\frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{b} + \frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{b} - \frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-2 \, b x - 2 \, d\right )} + 1\right )}{b} + \frac {2 \, {\left (e^{\left (-2 \, b x - 2 \, d\right )} + 2\right )} e^{\left (2 \, a - 2 \, d\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, d\right )} + e^{\left (-4 \, b x - 4 \, d\right )} + 1\right )}} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^3,x, algorithm="maxima")
Output:
e^(2*a - 2*d)*log(e^(-b*x - d) + 1)/b + e^(2*a - 2*d)*log(e^(-b*x - d) - 1 )/b - e^(2*a - 2*d)*log(e^(-2*b*x - 2*d) + 1)/b + 2*(e^(-2*b*x - 2*d) + 2) *e^(2*a - 2*d)/(b*(2*e^(-2*b*x - 2*d) + e^(-4*b*x - 4*d) + 1))
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=-\frac {{\left (2 \, e^{\left (-2 \, d\right )} \log \left (e^{\left (2 \, b x + 2 \, d\right )} + 1\right ) - 2 \, e^{\left (-2 \, d\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, d\right )} - 1 \right |}\right ) - \frac {{\left (3 \, e^{\left (4 \, b x + 4 \, d\right )} - 6 \, e^{\left (2 \, b x + 2 \, d\right )} - 5\right )} e^{\left (-2 \, d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} + 1\right )}^{2}}\right )} e^{\left (2 \, a\right )}}{2 \, b} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^3,x, algorithm="giac")
Output:
-1/2*(2*e^(-2*d)*log(e^(2*b*x + 2*d) + 1) - 2*e^(-2*d)*log(abs(e^(2*b*x + 2*d) - 1)) - (3*e^(4*b*x + 4*d) - 6*e^(2*b*x + 2*d) - 5)*e^(-2*d)/(e^(2*b* x + 2*d) + 1)^2)*e^(2*a)/b
Time = 2.68 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.40 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a-2\,d}}{b\,\left (2\,{\mathrm {e}}^{2\,d+2\,b\,x}+{\mathrm {e}}^{4\,d+4\,b\,x}+1\right )}-\frac {2\,\sqrt {{\mathrm {e}}^{4\,a-4\,d}}\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {-b^2}}{b\,\sqrt {{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}}}\right )}{\sqrt {-b^2}}-\frac {6\,{\mathrm {e}}^{2\,a-2\,d}}{b\,\left ({\mathrm {e}}^{2\,d+2\,b\,x}+1\right )} \] Input:
int(exp(2*a + 2*b*x)/(cosh(d + b*x)^3*sinh(d + b*x)),x)
Output:
(2*exp(2*a - 2*d))/(b*(2*exp(2*d + 2*b*x) + exp(4*d + 4*b*x) + 1)) - (2*ex p(4*a - 4*d)^(1/2)*atan((exp(2*a)*exp(2*b*x)*(-b^2)^(1/2))/(b*(exp(4*a)*ex p(-4*d))^(1/2))))/(-b^2)^(1/2) - (6*exp(2*a - 2*d))/(b*(exp(2*d + 2*b*x) + 1))
Time = 0.21 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.77 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^3(d+b x) \, dx=\frac {e^{2 a} \left (e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )+e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-e^{4 b x +4 d} \mathrm {log}\left (e^{2 b x +2 d}+1\right )+3 e^{4 b x +4 d}+2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )+2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )-2 e^{2 b x +2 d} \mathrm {log}\left (e^{2 b x +2 d}+1\right )+\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )-\mathrm {log}\left (e^{2 b x +2 d}+1\right )-1\right )}{e^{2 d} b \left (e^{4 b x +4 d}+2 e^{2 b x +2 d}+1\right )} \] Input:
int(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^3,x)
Output:
(e**(2*a)*(e**(4*b*x + 4*d)*log(e**(b*x + d) - 1) + e**(4*b*x + 4*d)*log(e **(b*x + d) + 1) - e**(4*b*x + 4*d)*log(e**(2*b*x + 2*d) + 1) + 3*e**(4*b* x + 4*d) + 2*e**(2*b*x + 2*d)*log(e**(b*x + d) - 1) + 2*e**(2*b*x + 2*d)*l og(e**(b*x + d) + 1) - 2*e**(2*b*x + 2*d)*log(e**(2*b*x + 2*d) + 1) + log( e**(b*x + d) - 1) + log(e**(b*x + d) + 1) - log(e**(2*b*x + 2*d) + 1) - 1) )/(e**(2*d)*b*(e**(4*b*x + 4*d) + 2*e**(2*b*x + 2*d) + 1))