Integrand size = 22, antiderivative size = 68 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {e^{a-2 d-b x}}{4 b}+\frac {e^{a+b x}}{b}+\frac {e^{a+2 d+3 b x}}{12 b}-\frac {2 e^{a-d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
1/4*exp(-b*x+a-2*d)/b+exp(b*x+a)/b+1/12*exp(3*b*x+a+2*d)/b-2*exp(a-d)*arct anh(exp(b*x+d))/b
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {e^{a-2 d-b x} \left (3+12 e^{2 (d+b x)}+e^{4 (d+b x)}-24 \sqrt {e^{2 (d+b x)}} \text {arctanh}\left (\sqrt {e^{2 (d+b x)}}\right )\right )}{12 b} \] Input:
Integrate[E^(a + b*x)*Cosh[d + b*x]^2*Coth[d + b*x],x]
Output:
(E^(a - 2*d - b*x)*(3 + 12*E^(2*(d + b*x)) + E^(4*(d + b*x)) - 24*Sqrt[E^( 2*(d + b*x))]*ArcTanh[Sqrt[E^(2*(d + b*x))]]))/(12*b)
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2720, 27, 364, 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.
\(\displaystyle \int e^{a+b x} \cosh ^2(b x+d) \coth (b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {e^{a-2 b x} \left (1+e^{2 b x}\right )^3}{4 \left (1-e^{2 b x}\right )}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {e^a \int \frac {e^{-2 b x} \left (1+e^{2 b x}\right )^3}{1-e^{2 b x}}de^{b x}}{4 b}\) |
\(\Big \downarrow \) 364 |
\(\displaystyle -\frac {e^a \int \left (e^{-2 b x}-e^{2 b x}-4-\frac {8}{-1+e^{2 b x}}\right )de^{b x}}{4 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^a \left (8 \text {arctanh}\left (e^{b x}\right )-e^{-b x}-4 e^{b x}-\frac {1}{3} e^{3 b x}\right )}{4 b}\) |
Input:
Int[E^(a + b*x)*Cosh[d + b*x]^2*Coth[d + b*x],x]
Output:
-1/4*(E^a*(-E^(-(b*x)) - 4*E^(b*x) - E^(3*b*x)/3 + 8*ArcTanh[E^(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[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
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 = 1.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {{\mathrm e}^{3 b x +a +2 d}}{12 b}+\frac {{\mathrm e}^{b x +a}}{b}+\frac {{\mathrm e}^{-b x +a -2 d}}{4 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}\) | \(93\) |
Input:
int(exp(b*x+a)*cosh(b*x+d)^2*coth(b*x+d),x,method=_RETURNVERBOSE)
Output:
1/12*exp(3*b*x+a+2*d)/b+exp(b*x+a)/b+1/4*exp(-b*x+a-2*d)/b+ln(exp(b*x+a)-e xp(a-d))/b*exp(a-d)-ln(exp(b*x+a)+exp(a-d))/b*exp(a-d)
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (59) = 118\).
Time = 0.08 (sec) , antiderivative size = 400, normalized size of antiderivative = 5.88 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {\cosh \left (b x + d\right )^{4} \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{4} + 4 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{3} + 12 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + 6 \, {\left (\cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) - {\left (\cosh \left (b x + d\right )^{2} + 2\right )} \sinh \left (-a + d\right ) + 2 \, \cosh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} - 12 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \log \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right ) + 1\right ) + 12 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \log \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right ) - 1\right ) + 4 \, {\left (\cosh \left (b x + d\right )^{3} \cosh \left (-a + d\right ) + 6 \, \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - {\left (\cosh \left (b x + d\right )^{3} + 6 \, \cosh \left (b x + d\right )\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{4} + 12 \, \cosh \left (b x + d\right )^{2} + 3\right )} \sinh \left (-a + d\right ) + 3 \, \cosh \left (-a + d\right )}{12 \, {\left (b \cosh \left (b x + d\right ) + b \sinh \left (b x + d\right )\right )}} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^2*coth(b*x+d),x, algorithm="fricas")
Output:
1/12*(cosh(b*x + d)^4*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b* x + d)^4 + 4*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sin h(b*x + d)^3 + 12*cosh(b*x + d)^2*cosh(-a + d) + 6*(cosh(b*x + d)^2*cosh(- a + d) - (cosh(b*x + d)^2 + 2)*sinh(-a + d) + 2*cosh(-a + d))*sinh(b*x + d )^2 - 12*(cosh(b*x + d)*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh( b*x + d) - cosh(b*x + d)*sinh(-a + d))*log(cosh(b*x + d) + sinh(b*x + d) + 1) + 12*(cosh(b*x + d)*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh( b*x + d) - cosh(b*x + d)*sinh(-a + d))*log(cosh(b*x + d) + sinh(b*x + d) - 1) + 4*(cosh(b*x + d)^3*cosh(-a + d) + 6*cosh(b*x + d)*cosh(-a + d) - (co sh(b*x + d)^3 + 6*cosh(b*x + d))*sinh(-a + d))*sinh(b*x + d) - (cosh(b*x + d)^4 + 12*cosh(b*x + d)^2 + 3)*sinh(-a + d) + 3*cosh(-a + d))/(b*cosh(b*x + d) + b*sinh(b*x + d))
\[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=e^{a} \int e^{b x} \cosh ^{2}{\left (b x + d \right )} \coth {\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)**2*coth(b*x+d),x)
Output:
exp(a)*Integral(exp(b*x)*cosh(b*x + d)**2*coth(b*x + d), x)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {{\left (e^{\left (3 \, b x + 3 \, a + 2 \, d\right )} + 12 \, e^{\left (b x + 3 \, a\right )}\right )} e^{\left (-2 \, a\right )}}{12 \, b} - \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{b} + \frac {e^{\left (-b x + a - 2 \, d\right )}}{4 \, b} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^2*coth(b*x+d),x, algorithm="maxima")
Output:
1/12*(e^(3*b*x + 3*a + 2*d) + 12*e^(b*x + 3*a))*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 + 1/4*e ^(-b*x + a - 2*d)/b
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.18 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {{\left (e^{\left (3 \, b x + a + 5 \, d\right )} + 12 \, e^{\left (b x + a + 3 \, d\right )}\right )} e^{\left (-3 \, d\right )} - 12 \, e^{\left (a - d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) + 12 \, e^{\left (a - d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) + 3 \, e^{\left (-b x + a - 2 \, d\right )}}{12 \, b} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^2*coth(b*x+d),x, algorithm="giac")
Output:
1/12*((e^(3*b*x + a + 5*d) + 12*e^(b*x + a + 3*d))*e^(-3*d) - 12*e^(a - d) *log(e^(b*x + d) + 1) + 12*e^(a - d)*log(abs(e^(b*x + d) - 1)) + 3*e^(-b*x + a - 2*d))/b
Time = 2.93 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.31 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {{\mathrm {e}}^{a-2\,d-b\,x}}{4\,b}+\frac {{\mathrm {e}}^{a+2\,d+3\,b\,x}}{12\,b}+\frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\sqrt {{\mathrm {e}}^{2\,a-2\,d}}\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b\,\sqrt {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,d}}}\right )}{\sqrt {-b^2}} \] Input:
int(cosh(d + b*x)^2*coth(d + b*x)*exp(a + b*x),x)
Output:
exp(a - 2*d - b*x)/(4*b) + exp(a + 2*d + 3*b*x)/(12*b) + exp(a + b*x)/b - (2*exp(2*a - 2*d)^(1/2)*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/(b*(exp(2*a)*e xp(-2*d))^(1/2))))/(-b^2)^(1/2)
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19 \[ \int e^{a+b x} \cosh ^2(d+b x) \coth (d+b x) \, dx=\frac {e^{a} \left (e^{4 b x +4 d}+12 e^{2 b x +2 d}+12 e^{b x +d} \mathrm {log}\left (e^{b x +d}-1\right )-12 e^{b x +d} \mathrm {log}\left (e^{b x +d}+1\right )+3\right )}{12 e^{b x +2 d} b} \] Input:
int(exp(b*x+a)*cosh(b*x+d)^2*coth(b*x+d),x)
Output:
(e**a*(e**(4*b*x + 4*d) + 12*e**(2*b*x + 2*d) + 12*e**(b*x + d)*log(e**(b* x + d) - 1) - 12*e**(b*x + d)*log(e**(b*x + d) + 1) + 3))/(12*e**(b*x + 2* d)*b)