Integrand size = 22, antiderivative size = 83 \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=-\frac {2 e^{a-d}}{b \left (1-e^{2 d+2 b x}\right )^2}+\frac {4 e^{a-d}}{b \left (1-e^{2 d+2 b x}\right )}+\frac {e^{a-d} \log \left (1-e^{2 d+2 b x}\right )}{b} \] 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 = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78 \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=\frac {2 e^{a-d} \left (-\frac {1}{\left (1-e^{2 (d+b x)}\right )^2}+\frac {2}{1-e^{2 (d+b x)}}+\frac {1}{2} \log \left (1-e^{2 (d+b x)}\right )\right )}{b} \] Input:
Integrate[E^(a + b*x)*Coth[d + b*x]^2*Csch[d + b*x],x]
Output:
(2*E^(a - d)*(-(1 - E^(2*(d + b*x)))^(-2) + 2/(1 - E^(2*(d + b*x))) + Log[ 1 - E^(2*(d + b*x))]/2))/b
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.60, 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.
\(\displaystyle \int e^{a+b x} \coth ^2(b x+d) \text {csch}(b x+d) \, dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle -\frac {e^a \int \left (-\frac {4}{\left (-1+e^{2 b x}\right )^2}-\frac {4}{\left (-1+e^{2 b x}\right )^3}+\frac {1}{1-e^{2 b x}}\right )de^{2 b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^a \left (-\frac {4}{1-e^{2 b x}}+\frac {2}{\left (1-e^{2 b x}\right )^2}-\log \left (1-e^{2 b x}\right )\right )}{b}\) |
Input:
Int[E^(a + b*x)*Coth[d + b*x]^2*Csch[d + b*x],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 = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.18
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{a -d} a}{b}+\frac {2 \left (-2 \,{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) {\mathrm e}^{3 a -d}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 d}\right ) {\mathrm e}^{a -d}}{b}\) | \(98\) |
Input:
int(exp(b*x+a)*coth(b*x+d)^2*csch(b*x+d),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. 487 vs. \(2 (73) = 146\).
Time = 0.09 (sec) , antiderivative size = 487, normalized size of antiderivative = 5.87 \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(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 \, \sinh \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)*coth(b*x+d)^2*csch(b*x+d),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))*s inh(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*sinh( 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)*si nh(-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)
\[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=e^{a} \int e^{b x} \coth ^{2}{\left (b x + d \right )} \operatorname {csch}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*coth(b*x+d)**2*csch(b*x+d),x)
Output:
exp(a)*Integral(exp(b*x)*coth(b*x + d)**2*csch(b*x + d), x)
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.28 \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=\frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{b} - \frac {2 \, {\left (2 \, e^{\left (2 \, b x + 5 \, a + 2 \, d\right )} - e^{\left (5 \, a\right )}\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a + 5 \, d\right )} - 2 \, e^{\left (2 \, b x + 4 \, a + 3 \, d\right )} + e^{\left (4 \, a + d\right )}\right )}} \] Input:
integrate(exp(b*x+a)*coth(b*x+d)^2*csch(b*x+d),x, algorithm="maxima")
Output:
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*(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 = 97, normalized size of antiderivative = 1.17 \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=\frac {2 \, e^{\left (a - d\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - e^{\left (2 \, a\right )} \right |}\right ) - \frac {{\left (3 \, e^{\left (4 \, b x + 5 \, a + 4 \, d\right )} + 2 \, e^{\left (2 \, b x + 5 \, a + 2 \, d\right )} - e^{\left (5 \, a\right )}\right )} e^{\left (-d\right )}}{{\left (e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - e^{\left (2 \, a\right )}\right )}^{2}}}{2 \, b} \] Input:
integrate(exp(b*x+a)*coth(b*x+d)^2*csch(b*x+d),x, algorithm="giac")
Output:
1/2*(2*e^(a - d)*log(abs(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. \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=\int \frac {{\mathrm {coth}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}}{\mathrm {sinh}\left (d+b\,x\right )} \,d x \] Input:
int((coth(d + b*x)^2*exp(a + b*x))/sinh(d + b*x),x)
Output:
int((coth(d + b*x)^2*exp(a + b*x))/sinh(d + b*x), x)
Time = 0.21 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.89 \[ \int e^{a+b x} \coth ^2(d+b x) \text {csch}(d+b x) \, dx=\frac {e^{a} \left (e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )+e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-2 e^{4 b x +4 d}-2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )+\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )\right )}{e^{d} b \left (e^{4 b x +4 d}-2 e^{2 b x +2 d}+1\right )} \] Input:
int(exp(b*x+a)*coth(b*x+d)^2*csch(b*x+d),x)
Output:
(e**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) - 2*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)*log(e**(b*x + d) + 1) + log(e**(b*x + d) - 1) + lo g(e**(b*x + d) + 1)))/(e**d*b*(e**(4*b*x + 4*d) - 2*e**(2*b*x + 2*d) + 1))