Integrand size = 22, antiderivative size = 77 \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=-\frac {2 e^{a+b x}}{b \left (1-e^{2 d+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1-e^{2 d+2 b x}\right )}-\frac {e^{a-d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
-2*exp(b*x+a)/b/(1-exp(2*b*x+2*d))^2+3*exp(b*x+a)/b/(1-exp(2*b*x+2*d))-exp (a-d)*arctanh(exp(b*x+d))/b
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\frac {e^a \left (\frac {e^{b x}-3 e^{2 d+3 b x}}{\left (-1+e^{2 (d+b x)}\right )^2}-e^{-d} \text {arctanh}\left (e^{d+b x}\right )\right )}{b} \] Input:
Integrate[E^(a + b*x)*Coth[d + b*x]*Csch[d + b*x]^2,x]
Output:
(E^a*((E^(b*x) - 3*E^(2*d + 3*b*x))/(-1 + E^(2*(d + b*x)))^2 - ArcTanh[E^( d + b*x)]/E^d))/b
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86, 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, 298, 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.
\(\displaystyle \int e^{a+b x} \coth (b x+d) \text {csch}^2(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {4 e^{a+2 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 e^a \int \frac {e^{2 b x} \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 360 |
\(\displaystyle -\frac {4 e^a \left (\frac {e^{b x}}{2 \left (1-e^{2 b x}\right )^2}-\frac {1}{4} \int \frac {2 \left (1+2 e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 e^a \left (\frac {e^{b x}}{2 \left (1-e^{2 b x}\right )^2}-\frac {1}{2} \int \frac {1+2 e^{2 b x}}{\left (1-e^{2 b x}\right )^2}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle -\frac {4 e^a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1-e^{2 b x}}de^{b x}-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{2 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {4 e^a \left (\frac {1}{2} \left (\frac {1}{2} \text {arctanh}\left (e^{b x}\right )-\frac {3 e^{b x}}{2 \left (1-e^{2 b x}\right )}\right )+\frac {e^{b x}}{2 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
Input:
Int[E^(a + b*x)*Coth[d + b*x]*Csch[d + b*x]^2,x]
Output:
(-4*E^a*(E^(b*x)/(2*(1 - E^(2*b*x))^2) + ((-3*E^(b*x))/(2*(1 - E^(2*b*x))) + ArcTanh[E^(b*x)]/2)/2))/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[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 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 = 0.60 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.38
method | result | size |
risch | \(\frac {\left (-3 \,{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) {\mathrm e}^{b x +3 a}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}\) | \(106\) |
Input:
int(exp(b*x+a)*coth(b*x+d)*csch(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/(-exp(2*b*x+2*a+2*d)+exp(2*a))^2/b*(-3*exp(2*b*x+2*a+2*d)+exp(2*a))*exp( b*x+3*a)+1/2*ln(exp(b*x+a)-exp(a-d))/b*exp(a-d)-1/2*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. 806 vs. \(2 (67) = 134\).
Time = 0.09 (sec) , antiderivative size = 806, normalized size of antiderivative = 10.47 \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(b*x+a)*coth(b*x+d)*csch(b*x+d)^2,x, algorithm="fricas")
Output:
-1/2*(6*cosh(b*x + d)^3*cosh(-a + d) + 6*(cosh(-a + d) - sinh(-a + d))*sin h(b*x + d)^3 + 18*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d) )*sinh(b*x + d)^2 - 2*cosh(b*x + d)*cosh(-a + d) + (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))*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*co sh(-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(cosh(b*x + d) + sinh(b*x + d) + 1) - (cos h(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))*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*co sh(b*x + d)^2 + 1)*sinh(-a + d) + cosh(-a + d))*log(cosh(b*x + d) + sinh(b *x + d) - 1) + 2*(9*cosh(b*x + d)^2*cosh(-a + d) - (9*cosh(b*x + d)^2 - 1) *sinh(-a + d) - cosh(-a + d))*sinh(b*x + d) - 2*(3*cosh(b*x + d)^3 - cosh( b*x + d))*sinh(-a + d))/(b*cosh(b*x + d)^4 + 4*b*cosh(b*x + d)*sinh(b*x...
\[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=e^{a} \int e^{b x} \coth {\left (b x + d \right )} \operatorname {csch}^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*coth(b*x+d)*csch(b*x+d)**2,x)
Output:
exp(a)*Integral(exp(b*x)*coth(b*x + d)*csch(b*x + d)**2, x)
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.43 \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=-\frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{2 \, b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{2 \, b} - \frac {3 \, e^{\left (3 \, b x + 5 \, a + 2 \, d\right )} - e^{\left (b x + 5 \, a\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a + 4 \, d\right )} - 2 \, e^{\left (2 \, b x + 4 \, a + 2 \, d\right )} + e^{\left (4 \, a\right )}\right )}} \] Input:
integrate(exp(b*x+a)*coth(b*x+d)*csch(b*x+d)^2,x, algorithm="maxima")
Output:
-1/2*e^(a - d)*log(e^(b*x + a + d) + e^a)/b + 1/2*e^(a - d)*log(e^(b*x + a + d) - e^a)/b - (3*e^(3*b*x + 5*a + 2*d) - e^(b*x + 5*a))/(b*(e^(4*b*x + 4*a + 4*d) - 2*e^(2*b*x + 4*a + 2*d) + e^(4*a)))
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=-\frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) - e^{\left (a - d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) + \frac {2 \, {\left (3 \, e^{\left (3 \, b x + a + 3 \, d\right )} - e^{\left (b x + a + d\right )}\right )} e^{\left (-d\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{2}}}{2 \, b} \] Input:
integrate(exp(b*x+a)*coth(b*x+d)*csch(b*x+d)^2,x, algorithm="giac")
Output:
-1/2*(e^(a - d)*log(e^(b*x + d) + 1) - e^(a - d)*log(abs(e^(b*x + d) - 1)) + 2*(3*e^(3*b*x + a + 3*d) - e^(b*x + a + d))*e^(-d)/(e^(2*b*x + 2*d) - 1 )^2)/b
Timed out. \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\int \frac {\mathrm {coth}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x}}{{\mathrm {sinh}\left (d+b\,x\right )}^2} \,d x \] Input:
int((coth(d + b*x)*exp(a + b*x))/sinh(d + b*x)^2,x)
Output:
int((coth(d + b*x)*exp(a + b*x))/sinh(d + b*x)^2, x)
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.21 \[ \int e^{a+b x} \coth (d+b x) \text {csch}^2(d+b x) \, dx=\frac {e^{a} \left (e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )-e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-6 e^{3 b x +3 d}-2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )+2 e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )+2 e^{b x +d}+\mathrm {log}\left (e^{b x +d}-1\right )-\mathrm {log}\left (e^{b x +d}+1\right )\right )}{2 e^{d} b \left (e^{4 b x +4 d}-2 e^{2 b x +2 d}+1\right )} \] Input:
int(exp(b*x+a)*coth(b*x+d)*csch(b*x+d)^2,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) - 6*e**(3*b*x + 3*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) + 2*e**(b*x + d) + log(e**(b *x + d) - 1) - log(e**(b*x + d) + 1)))/(2*e**d*b*(e**(4*b*x + 4*d) - 2*e** (2*b*x + 2*d) + 1))