Integrand size = 22, antiderivative size = 73 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=\frac {2 e^{2 a-d+b x}}{b}+\frac {2 e^{2 a-d+b x}}{b \left (1-e^{2 d+2 b x}\right )}-\frac {4 e^{2 a-2 d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
2*exp(b*x+2*a-d)/b+2*exp(b*x+2*a-d)/b/(1-exp(2*b*x+2*d))-4*exp(2*a-2*d)*ar ctanh(exp(b*x+d))/b
Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(73)=146\).
Time = 0.46 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.89 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=\frac {2 e^{2 a} \left (e^{b x} \cosh (d)-\cosh (2 d) \log \left (\left (1+e^{b x}\right ) \cosh \left (\frac {d}{2}\right )+\left (-1+e^{b x}\right ) \sinh \left (\frac {d}{2}\right )\right )+\cosh (2 d) \log \left (\left (-1+e^{b x}\right ) \cosh \left (\frac {d}{2}\right )+\left (1+e^{b x}\right ) \sinh \left (\frac {d}{2}\right )\right )-e^{b x} \sinh (d)-\frac {e^{b x} (\cosh (d)-\sinh (d))^2}{\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)}+\log \left (\left (1+e^{b x}\right ) \cosh \left (\frac {d}{2}\right )+\left (-1+e^{b x}\right ) \sinh \left (\frac {d}{2}\right )\right ) \sinh (2 d)-\log \left (\left (-1+e^{b x}\right ) \cosh \left (\frac {d}{2}\right )+\left (1+e^{b x}\right ) \sinh \left (\frac {d}{2}\right )\right ) \sinh (2 d)\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))*Coth[d + b*x]*Csch[d + b*x],x]
Output:
(2*E^(2*a)*(E^(b*x)*Cosh[d] - Cosh[2*d]*Log[(1 + E^(b*x))*Cosh[d/2] + (-1 + E^(b*x))*Sinh[d/2]] + Cosh[2*d]*Log[(-1 + E^(b*x))*Cosh[d/2] + (1 + E^(b *x))*Sinh[d/2]] - E^(b*x)*Sinh[d] - (E^(b*x)*(Cosh[d] - Sinh[d])^2)/((-1 + E^(2*b*x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d]) + Log[(1 + E^(b*x))*Cosh[d/ 2] + (-1 + E^(b*x))*Sinh[d/2]]*Sinh[2*d] - Log[(-1 + E^(b*x))*Cosh[d/2] + (1 + E^(b*x))*Sinh[d/2]]*Sinh[2*d]))/b
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.58, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2720, 27, 360, 27, 299, 219}
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.
\(\displaystyle \int e^{2 (a+b x)} \coth (b x+d) \text {csch}(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {2 e^{2 a+2 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 e^{2 a} \int \frac {e^{2 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 360 |
\(\displaystyle \frac {2 e^{2 a} \left (\frac {e^{b x}}{1-e^{2 b x}}-\frac {1}{2} \int \frac {2 \left (1+e^{2 b x}\right )}{1-e^{2 b x}}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 e^{2 a} \left (\frac {e^{b x}}{1-e^{2 b x}}-\int \frac {1+e^{2 b x}}{1-e^{2 b x}}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {2 e^{2 a} \left (-2 \int \frac {1}{1-e^{2 b x}}de^{b x}+e^{b x}+\frac {e^{b x}}{1-e^{2 b x}}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 e^{2 a} \left (-2 \text {arctanh}\left (e^{b x}\right )+e^{b x}+\frac {e^{b x}}{1-e^{2 b x}}\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))*Coth[d + b*x]*Csch[d + b*x],x]
Output:
(2*E^(2*a)*(E^(b*x) + E^(b*x)/(1 - E^(2*b*x)) - 2*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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 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 = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +2 a -d}}{b}+\frac {2 \,{\mathrm e}^{b x +4 a -d}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) b}+\frac {2 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}\) | \(111\) |
Input:
int(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d),x,method=_RETURNVERBOSE)
Output:
2*exp(b*x+2*a-d)/b+2/(-exp(2*b*x+2*a+2*d)+exp(2*a))/b*exp(b*x+4*a-d)+2*ln( exp(b*x+a)-exp(a-d))/b*exp(2*a-2*d)-2*ln(exp(b*x+a)+exp(a-d))/b*exp(2*a-2* d)
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (66) = 132\).
Time = 0.09 (sec) , antiderivative size = 497, normalized size of antiderivative = 6.81 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d),x, algorithm="fricas")
Output:
2*(cosh(b*x + d)^3*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d) )*sinh(b*x + d)^3 + 3*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh (-2*a + 2*d))*sinh(b*x + d)^2 - 2*cosh(b*x + d)*cosh(-2*a + 2*d) - (cosh(b *x + d)^2*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b* x + d)^2 + 2*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2 *d))*sinh(b*x + d) - (cosh(b*x + d)^2 - 1)*sinh(-2*a + 2*d) - cosh(-2*a + 2*d))*log(cosh(b*x + d) + sinh(b*x + d) + 1) + (cosh(b*x + d)^2*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d)^2 + 2*(cosh(b *x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh(b*x + d) - (cosh(b*x + d)^2 - 1)*sinh(-2*a + 2*d) - cosh(-2*a + 2*d))*log(cosh(b*x + d) + sinh(b*x + d) - 1) + (3*cosh(b*x + d)^2*cosh(-2*a + 2*d) - (3*cosh(b *x + d)^2 - 2)*sinh(-2*a + 2*d) - 2*cosh(-2*a + 2*d))*sinh(b*x + d) - (cos h(b*x + d)^3 - 2*cosh(b*x + d))*sinh(-2*a + 2*d))/(b*cosh(b*x + d)^2 + 2*b *cosh(b*x + d)*sinh(b*x + d) + b*sinh(b*x + d)^2 - b)
\[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=e^{2 a} \int e^{2 b x} \coth {\left (b x + d \right )} \operatorname {csch}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d),x)
Output:
exp(2*a)*Integral(exp(2*b*x)*coth(b*x + d)*csch(b*x + d), x)
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.37 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=-\frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{b} + \frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{b} - \frac {2 \, {\left (2 \, e^{\left (-2 \, b x - 2 \, d\right )} - 1\right )} e^{\left (2 \, a - 2 \, d\right )}}{b {\left (e^{\left (-b x - d\right )} - e^{\left (-3 \, b x - 3 \, d\right )}\right )}} \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d),x, algorithm="maxima")
Output:
-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*(2*e^(-2*b*x - 2*d) - 1)*e^(2*a - 2*d)/(b*(e^(-b*x - d) - e^( -3*b*x - 3*d)))
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.12 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=-\frac {2 \, {\left (e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) - e^{\left (2 \, a - 2 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) + \frac {e^{\left (b x + 2 \, a - d\right )}}{e^{\left (2 \, b x + 2 \, d\right )} - 1} - e^{\left (b x + 2 \, a - d\right )}\right )}}{b} \] Input:
integrate(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d),x, algorithm="giac")
Output:
-2*(e^(2*a - 2*d)*log(e^(b*x + d) + 1) - e^(2*a - 2*d)*log(abs(e^(b*x + d) - 1)) + e^(b*x + 2*a - d)/(e^(2*b*x + 2*d) - 1) - e^(b*x + 2*a - d))/b
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.37 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b}-\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b\,\left ({\mathrm {e}}^{2\,d+2\,b\,x}-1\right )}-\frac {4\,\sqrt {{\mathrm {e}}^{4\,a-4\,d}}\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-d}\,{\mathrm {e}}^{b\,x}\,\sqrt {-b^2}}{b\,\sqrt {{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}}}\right )}{\sqrt {-b^2}} \] Input:
int((coth(d + b*x)*exp(2*a + 2*b*x))/sinh(d + b*x),x)
Output:
(2*exp(2*a - d + b*x))/b - (2*exp(2*a - d + b*x))/(b*(exp(2*d + 2*b*x) - 1 )) - (4*exp(4*a - 4*d)^(1/2)*atan((exp(2*a)*exp(-d)*exp(b*x)*(-b^2)^(1/2)) /(b*(exp(4*a)*exp(-4*d))^(1/2))))/(-b^2)^(1/2)
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59 \[ \int e^{2 (a+b x)} \coth (d+b x) \text {csch}(d+b x) \, dx=\frac {2 e^{2 a} \left (e^{3 b x +3 d}+e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )-2 e^{b x +d}-\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )\right )}{e^{2 d} b \left (e^{2 b x +2 d}-1\right )} \] Input:
int(exp(2*b*x+2*a)*coth(b*x+d)*csch(b*x+d),x)
Output:
(2*e**(2*a)*(e**(3*b*x + 3*d) + e**(2*b*x + 2*d)*log(e**(b*x + d) - 1) - e **(2*b*x + 2*d)*log(e**(b*x + d) + 1) - 2*e**(b*x + d) - log(e**(b*x + d) - 1) + log(e**(b*x + d) + 1)))/(e**(2*d)*b*(e**(2*b*x + 2*d) - 1))