Integrand size = 16, antiderivative size = 49 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=\frac {2 e^{a+b x}}{b \left (1-e^{2 d+2 b x}\right )}-\frac {2 e^{a-d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
2*exp(b*x+a)/b/(1-exp(2*b*x+2*d))-2*exp(a-d)*arctanh(exp(b*x+d))/b
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=-\frac {2 e^a (\cosh (d)-\sinh (d)) \left (e^{b x}+\text {arctanh}\left (e^{b x} (\cosh (d)+\sinh (d))\right ) \left (\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)\right )\right )}{b \left (\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)\right )} \] Input:
Integrate[E^(a + b*x)*Csch[d + b*x]^2,x]
Output:
(-2*E^a*(Cosh[d] - Sinh[d])*(E^(b*x) + ArcTanh[E^(b*x)*(Cosh[d] + Sinh[d]) ]*((-1 + E^(2*b*x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d])))/(b*((-1 + E^(2*b* x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d]))
Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2720, 27, 252, 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.
\(\displaystyle \int e^{a+b x} \text {csch}^2(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {4 e^{a+2 b x}}{\left (1-e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 e^a \int \frac {e^{2 b x}}{\left (1-e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {4 e^a \left (\frac {e^{b x}}{2 \left (1-e^{2 b x}\right )}-\frac {1}{2} \int \frac {1}{1-e^{2 b x}}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {4 e^a \left (\frac {e^{b x}}{2 \left (1-e^{2 b x}\right )}-\frac {1}{2} \text {arctanh}\left (e^{b x}\right )\right )}{b}\) |
Input:
Int[E^(a + b*x)*Csch[d + b*x]^2,x]
Output:
(4*E^a*(E^(b*x)/(2*(1 - E^(2*b*x))) - ArcTanh[E^(b*x)]/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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
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 = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.78
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +3 a}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}\) | \(87\) |
Input:
int(exp(b*x+a)*csch(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
2/(-exp(2*b*x+2*a+2*d)+exp(2*a))/b*exp(b*x+3*a)+ln(exp(b*x+a)-exp(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. 332 vs. \(2 (43) = 86\).
Time = 0.09 (sec) , antiderivative size = 332, normalized size of antiderivative = 6.78 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=-\frac {2 \, \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) + {\left (\cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} + 2 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{2} - 1\right )} \sinh \left (-a + d\right ) - \cosh \left (-a + d\right )\right )} \log \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right ) + 1\right ) - {\left (\cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} + 2 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{2} - 1\right )} \sinh \left (-a + d\right ) - \cosh \left (-a + d\right )\right )} \log \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right ) - 1\right ) + 2 \, {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - 2 \, \cosh \left (b x + d\right ) \sinh \left (-a + d\right )}{b \cosh \left (b x + d\right )^{2} + 2 \, b \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + b \sinh \left (b x + d\right )^{2} - b} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)^2,x, algorithm="fricas")
Output:
-(2*cosh(b*x + d)*cosh(-a + d) + (cosh(b*x + d)^2*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^2 + 2*(cosh(b*x + d)*cosh(-a + d) - cos h(b*x + d)*sinh(-a + d))*sinh(b*x + d) - (cosh(b*x + d)^2 - 1)*sinh(-a + d ) - cosh(-a + d))*log(cosh(b*x + d) + sinh(b*x + d) + 1) - (cosh(b*x + d)^ 2*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^2 + 2*(cosh(b *x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d) - (cosh(b *x + d)^2 - 1)*sinh(-a + d) - cosh(-a + d))*log(cosh(b*x + d) + sinh(b*x + d) - 1) + 2*(cosh(-a + d) - sinh(-a + d))*sinh(b*x + d) - 2*cosh(b*x + d) *sinh(-a + d))/(b*cosh(b*x + d)^2 + 2*b*cosh(b*x + d)*sinh(b*x + d) + b*si nh(b*x + d)^2 - b)
\[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*csch(b*x+d)**2,x)
Output:
exp(a)*Integral(exp(b*x)*csch(b*x + d)**2, x)
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.63 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=-\frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{b} - \frac {2 \, e^{\left (b x + 3 \, a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - e^{\left (2 \, a\right )}\right )}} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)^2,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*e^(b*x + 3*a)/(b*(e^(2*b*x + 2*a + 2*d) - e^(2*a)))
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.45 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=-{\left (\frac {e^{\left (-3 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right )}{b} - \frac {e^{\left (-3 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right )}{b} + \frac {2 \, e^{\left (b x - 2 \, d\right )}}{b {\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}}\right )} e^{\left (a + 2 \, d\right )} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)^2,x, algorithm="giac")
Output:
-(e^(-3*d)*log(e^(b*x + d) + 1)/b - e^(-3*d)*log(abs(e^(b*x + d) - 1))/b + 2*e^(b*x - 2*d)/(b*(e^(2*b*x + 2*d) - 1)))*e^(a + 2*d)
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.82 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{3\,a}\,{\mathrm {e}}^{-2\,d}\,{\mathrm {e}}^{b\,x}}{b\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,d}-b\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b\,\sqrt {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,d}}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,d}}}{\sqrt {-b^2}} \] Input:
int(exp(a + b*x)/sinh(d + b*x)^2,x)
Output:
(2*exp(3*a)*exp(-2*d)*exp(b*x))/(b*exp(2*a)*exp(-2*d) - b*exp(2*a)*exp(2*b *x)) - (2*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/(b*(exp(2*a)*exp(-2*d))^(1/2 )))*(exp(2*a)*exp(-2*d))^(1/2))/(-b^2)^(1/2)
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.06 \[ \int e^{a+b x} \text {csch}^2(d+b x) \, dx=\frac {e^{a} \left (e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )-2 e^{b x +d}-\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )\right )}{e^{d} b \left (e^{2 b x +2 d}-1\right )} \] Input:
int(exp(b*x+a)*csch(b*x+d)^2,x)
Output:
(e**a*(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**d*b*(e**(2*b*x + 2*d) - 1))