Integrand size = 20, antiderivative size = 41 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=\frac {2 e^{a-d} \arctan \left (e^{d+b x}\right )}{b}-\frac {2 e^{a-d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
2*exp(a-d)*arctan(exp(b*x+d))/b-2*exp(a-d)*arctanh(exp(b*x+d))/b
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=\frac {2 e^a \left (\arctan \left (e^{b x} (\cosh (d)+\sinh (d))\right )-\text {arctanh}\left (e^{b x} (\cosh (d)+\sinh (d))\right )\right ) (\cosh (d)-\sinh (d))}{b} \] Input:
Integrate[E^(a + b*x)*Csch[d + b*x]*Sech[d + b*x],x]
Output:
(2*E^a*(ArcTan[E^(b*x)*(Cosh[d] + Sinh[d])] - ArcTanh[E^(b*x)*(Cosh[d] + S inh[d])])*(Cosh[d] - Sinh[d]))/b
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2720, 27, 827, 216, 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.
\(\displaystyle \int e^{a+b x} \text {csch}(b x+d) \text {sech}(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {4 e^{a+2 b x}}{1-e^{4 b x}}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 e^a \int \frac {e^{2 b x}}{1-e^{4 b x}}de^{b x}}{b}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle -\frac {4 e^a \left (\frac {1}{2} \int \frac {1}{1-e^{2 b x}}de^{b x}-\frac {1}{2} \int \frac {1}{1+e^{2 b x}}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {4 e^a \left (\frac {1}{2} \int \frac {1}{1-e^{2 b x}}de^{b x}-\frac {1}{2} \arctan \left (e^{b x}\right )\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {4 e^a \left (\frac {1}{2} \text {arctanh}\left (e^{b x}\right )-\frac {1}{2} \arctan \left (e^{b x}\right )\right )}{b}\) |
Input:
Int[E^(a + b*x)*Csch[d + b*x]*Sech[d + b*x],x]
Output:
(-4*E^a*(-1/2*ArcTan[E^(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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 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]]]
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.71
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{b}\) | \(111\) |
Input:
int(exp(b*x+a)*csch(b*x+d)*sech(b*x+d),x,method=_RETURNVERBOSE)
Output:
ln(exp(b*x+a)-exp(a-d))/b*exp(a-d)-ln(exp(b*x+a)+exp(a-d))/b*exp(a-d)+I*ln (exp(b*x+a)+I*exp(a-d))/b*exp(a-d)-I*ln(exp(b*x+a)-I*exp(a-d))/b*exp(a-d)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (37) = 74\).
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.41 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=\frac {2 \, {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \arctan \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right )\right ) - {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \log \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right ) + 1\right ) + {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \log \left (\cosh \left (b x + d\right ) + \sinh \left (b x + d\right ) - 1\right )}{b} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d),x, algorithm="fricas")
Output:
(2*(cosh(-a + d) - sinh(-a + d))*arctan(cosh(b*x + d) + sinh(b*x + d)) - ( cosh(-a + d) - sinh(-a + d))*log(cosh(b*x + d) + sinh(b*x + d) + 1) + (cos h(-a + d) - sinh(-a + d))*log(cosh(b*x + d) + sinh(b*x + d) - 1))/b
\[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}{\left (b x + d \right )} \operatorname {sech}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d),x)
Output:
exp(a)*Integral(exp(b*x)*csch(b*x + d)*sech(b*x + d), x)
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.56 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=\frac {2 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (a - d\right )}}{b} - \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{b} + \frac {e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{b} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d),x, algorithm="maxima")
Output:
2*arctan(e^(b*x + d))*e^(a - d)/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
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.46 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx={\left (\frac {2 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-3 \, d\right )}}{b} - \frac {e^{\left (-3 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right )}{b} + \frac {e^{\left (-3 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right )}{b}\right )} e^{\left (a + 2 \, d\right )} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)*sech(b*x+d),x, algorithm="giac")
Output:
(2*arctan(e^(b*x + d))*e^(-3*d)/b - e^(-3*d)*log(e^(b*x + d) + 1)/b + e^(- 3*d)*log(abs(e^(b*x + d) - 1))/b)*e^(a + 2*d)
Time = 4.98 (sec) , antiderivative size = 181, normalized size of antiderivative = 4.41 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=-\frac {{\left ({\mathrm {e}}^{4\,a-4\,d}\right )}^{1/4}\,\left (\ln \left ({\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-12\,d}+{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-10\,d}\,{\mathrm {e}}^{b\,x}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}\right )}^{1/4}\right )-\ln \left ({\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-12\,d}-{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-10\,d}\,{\mathrm {e}}^{b\,x}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}\right )}^{1/4}\right )-\ln \left ({\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-10\,d}\,{\mathrm {e}}^{b\,x}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}\right )}^{1/4}+{\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-12\,d}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left (64\,{\mathrm {e}}^{11\,a}\,{\mathrm {e}}^{-10\,d}\,{\mathrm {e}}^{b\,x}\,\sqrt {{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}}-{\mathrm {e}}^{12\,a}\,{\mathrm {e}}^{-12\,d}\,{\left ({\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-4\,d}\right )}^{1/4}\,64{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{b} \] Input:
int(exp(a + b*x)/(cosh(d + b*x)*sinh(d + b*x)),x)
Output:
-(exp(4*a - 4*d)^(1/4)*(log(exp(12*a)*exp(-12*d) + exp(11*a)*exp(-10*d)*ex p(b*x)*(exp(4*a)*exp(-4*d))^(1/4)) - log(exp(12*a)*exp(-12*d) - exp(11*a)* exp(-10*d)*exp(b*x)*(exp(4*a)*exp(-4*d))^(1/4)) - log(exp(12*a)*exp(-12*d) *1i + exp(11*a)*exp(-10*d)*exp(b*x)*(exp(4*a)*exp(-4*d))^(1/4))*1i + log(6 4*exp(11*a)*exp(-10*d)*exp(b*x)*(exp(4*a)*exp(-4*d))^(1/2) - exp(12*a)*exp (-12*d)*(exp(4*a)*exp(-4*d))^(1/4)*64i)*1i))/b
Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10 \[ \int e^{a+b x} \text {csch}(d+b x) \text {sech}(d+b x) \, dx=\frac {e^{a} \left (2 \mathit {atan} \left (e^{b x +d}\right )+\mathrm {log}\left (e^{b x +d}-1\right )-\mathrm {log}\left (e^{b x +d}+1\right )\right )}{e^{d} b} \] Input:
int(exp(b*x+a)*csch(b*x+d)*sech(b*x+d),x)
Output:
(e**a*(2*atan(e**(b*x + d)) + log(e**(b*x + d) - 1) - log(e**(b*x + d) + 1 )))/(e**d*b)