Integrand size = 24, antiderivative size = 90 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=-\frac {2 e^{2 a-2 d}}{b \left (1-e^{2 d+2 b x}\right )^2}+\frac {6 e^{2 a-2 d}}{b \left (1-e^{2 d+2 b x}\right )}+\frac {2 e^{2 a-2 d} \log \left (1-e^{2 d+2 b x}\right )}{b} \] 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)*ln(1-exp(2*b*x+2*d))/b
Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\frac {2 e^{2 a-2 d} \left (\frac {2-3 e^{2 (d+b x)}}{\left (-1+e^{2 (d+b x)}\right )^2}+\log \left (1-e^{2 (d+b x)}\right )\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))*Coth[d + b*x]*Csch[d + b*x]^2,x]
Output:
(2*E^(2*a - 2*d)*((2 - 3*E^(2*(d + b*x)))/(-1 + E^(2*(d + b*x)))^2 + Log[1 - E^(2*(d + b*x))]))/b
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, 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.
\(\displaystyle \int e^{2 (a+b x)} \coth (b x+d) \text {csch}^2(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {4 e^{2 a+3 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 e^{2 a} \int \frac {e^{3 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {2 e^{2 a} \int \frac {e^{2 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^3}de^{2 b x}}{b}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {2 e^{2 a} \int \left (-\frac {3}{\left (-1+e^{2 b x}\right )^2}-\frac {2}{\left (-1+e^{2 b x}\right )^3}+\frac {1}{1-e^{2 b x}}\right )de^{2 b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 e^{2 a} \left (-\frac {3}{1-e^{2 b x}}+\frac {1}{\left (e^{2 b x}-1\right )^2}-\log \left (1-e^{2 b x}\right )\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))*Coth[d + b*x]*Csch[d + b*x]^2,x]
Output:
(-2*E^(2*a)*(-3/(1 - E^(2*b*x)) + (-1 + E^(2*b*x))^(-2) - 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_))*((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]]]
Time = 0.57 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17
method | result | size |
risch | \(-\frac {4 \,{\mathrm e}^{2 a -2 d} a}{b}+\frac {2 \left (-3 \,{\mathrm e}^{2 b x +2 a +2 d}+2 \,{\mathrm e}^{2 a}\right ) {\mathrm e}^{4 a -2 d}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{2 a -2 d}}{b}\) | \(105\) |
Input:
int(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-4/b*exp(2*a-2*d)*a+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)+2*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. 529 vs. \(2 (80) = 160\).
Time = 0.09 (sec) , antiderivative size = 529, normalized size of antiderivative = 5.88 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d)^2,x, algorithm="fricas")
Output:
-2*(3*cosh(b*x + d)^2*cosh(-2*a + 2*d) + 3*(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*c osh(-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))*log(2*sinh(b*x + d)/(cosh(b*x + d) - sinh(b*x + d))) + 6*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d) - (3*cosh(b*x + d)^2 - 2)* sinh(-2*a + 2*d) - 2*cosh(-2*a + 2*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*cos h(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)
\[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=e^{2 a} \int e^{2 b x} \coth {\left (b x + d \right )} \operatorname {csch}^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d)**2,x)
Output:
exp(2*a)*Integral(exp(2*b*x)*coth(b*x + d)*csch(b*x + d)**2, x)
Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.40 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=4 \, x e^{\left (2 \, a - 2 \, d\right )} + \frac {4 \, d e^{\left (2 \, a - 2 \, d\right )}}{b} + \frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{b} + \frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{b} - \frac {2 \, {\left (e^{\left (-2 \, b x - 2 \, d\right )} - 2\right )} e^{\left (2 \, a - 2 \, d\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, d\right )} - e^{\left (-4 \, b x - 4 \, d\right )} - 1\right )}} \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d)^2,x, algorithm="maxima")
Output:
4*x*e^(2*a - 2*d) + 4*d*e^(2*a - 2*d)/b + 2*e^(2*a - 2*d)*log(e^(-b*x - d) + 1)/b + 2*e^(2*a - 2*d)*log(e^(-b*x - 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 = 68, normalized size of antiderivative = 0.76 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, d\right )} - 1 \right |}\right ) - \frac {{\left (3 \, e^{\left (4 \, b x + 2 \, a + 4 \, d\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{2}}}{b} \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d)^2,x, algorithm="giac")
Output:
(2*e^(2*a - 2*d)*log(abs(e^(2*b*x + 2*d) - 1)) - (3*e^(4*b*x + 2*a + 4*d) - e^(2*a))*e^(-2*d)/(e^(2*b*x + 2*d) - 1)^2)/b
Time = 2.98 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a-2\,d}\,\ln \left ({\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-\frac {2\,{\mathrm {e}}^{2\,a-2\,d}}{b\,\left ({\mathrm {e}}^{4\,d+4\,b\,x}-2\,{\mathrm {e}}^{2\,d+2\,b\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{2\,a-2\,d}}{b\,\left ({\mathrm {e}}^{2\,d+2\,b\,x}-1\right )} \] Input:
int((coth(d + b*x)*exp(2*a + 2*b*x))/sinh(d + b*x)^2,x)
Output:
(2*exp(2*a - 2*d)*log(exp(2*d)*exp(2*b*x) - 1))/b - (2*exp(2*a - 2*d))/(b* (exp(4*d + 4*b*x) - 2*exp(2*d + 2*b*x) + 1)) - (6*exp(2*a - 2*d))/(b*(exp( 2*d + 2*b*x) - 1))
Time = 0.24 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.87 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\frac {e^{2 a} \left (2 e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )+2 e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-3 e^{4 b x +4 d}-4 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-4 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )+2 \,\mathrm {log}\left (e^{b x +d}-1\right )+2 \,\mathrm {log}\left (e^{b x +d}+1\right )+1\right )}{e^{2 d} b \left (e^{4 b x +4 d}-2 e^{2 b x +2 d}+1\right )} \] Input:
int(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d)^2,x)
Output:
(e**(2*a)*(2*e**(4*b*x + 4*d)*log(e**(b*x + d) - 1) + 2*e**(4*b*x + 4*d)*l og(e**(b*x + d) + 1) - 3*e**(4*b*x + 4*d) - 4*e**(2*b*x + 2*d)*log(e**(b*x + d) - 1) - 4*e**(2*b*x + 2*d)*log(e**(b*x + d) + 1) + 2*log(e**(b*x + d) - 1) + 2*log(e**(b*x + d) + 1) + 1))/(e**(2*d)*b*(e**(4*b*x + 4*d) - 2*e* *(2*b*x + 2*d) + 1))