Integrand size = 24, antiderivative size = 76 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {2 e^{2 a-d+b x}}{b \left (1+e^{2 d+2 b x}\right )}+\frac {4 e^{2 a-2 d} \arctan \left (e^{d+b x}\right )}{b}-\frac {2 e^{2 a-2 d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
-2*exp(b*x+2*a-d)/b/(1+exp(2*b*x+2*d))+4*exp(2*a-2*d)*arctan(exp(b*x+d))/b -2*exp(2*a-2*d)*arctanh(exp(b*x+d))/b
Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.92 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\frac {e^{2 a-2 d} \left (-\frac {2 e^{d+b x}}{1+e^{2 (d+b x)}}+4 \arctan \left (e^{d+b x}\right )+\log \left (1-e^{d+b x}\right )-\log \left (1+e^{d+b x}\right )\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))*Csch[d + b*x]*Sech[d + b*x]^2,x]
Output:
(E^(2*a - 2*d)*((-2*E^(d + b*x))/(1 + E^(2*(d + b*x))) + 4*ArcTan[E^(d + b *x)] + Log[1 - E^(d + b*x)] - Log[1 + E^(d + b*x)]))/b
Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2720, 27, 372, 397, 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.
\(\displaystyle \int e^{2 (a+b x)} \text {csch}(b x+d) \text {sech}^2(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {8 e^{2 a+4 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {8 e^{2 a} \int \frac {e^{4 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )^2}de^{b x}}{b}\) |
\(\Big \downarrow \) 372 |
\(\displaystyle -\frac {8 e^{2 a} \left (\frac {e^{b x}}{4 \left (e^{2 b x}+1\right )}-\frac {1}{4} \int \frac {1-3 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle -\frac {8 e^{2 a} \left (\frac {1}{4} \left (\int \frac {1}{1-e^{2 b x}}de^{b x}-2 \int \frac {1}{1+e^{2 b x}}de^{b x}\right )+\frac {e^{b x}}{4 \left (e^{2 b x}+1\right )}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {8 e^{2 a} \left (\frac {1}{4} \left (\int \frac {1}{1-e^{2 b x}}de^{b x}-2 \arctan \left (e^{b x}\right )\right )+\frac {e^{b x}}{4 \left (e^{2 b x}+1\right )}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {8 e^{2 a} \left (\frac {1}{4} \left (\text {arctanh}\left (e^{b x}\right )-2 \arctan \left (e^{b x}\right )\right )+\frac {e^{b x}}{4 \left (e^{2 b x}+1\right )}\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))*Csch[d + b*x]*Sech[d + b*x]^2,x]
Output:
(-8*E^(2*a)*(E^(b*x)/(4*(1 + E^(2*b*x))) + (-2*ArcTan[E^(b*x)] + ArcTanh[E ^(b*x)])/4))/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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, 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]]]
Result contains complex when optimal does not.
Time = 13.74 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.03
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{b x +4 a -d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) b}+\frac {2 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {2 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}\) | \(154\) |
Input:
int(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-2/(exp(2*b*x+2*a+2*d)+exp(2*a))/b*exp(b*x+4*a-d)+2*I*ln(exp(b*x+a)+I*exp( a-d))/b*exp(2*a-2*d)-2*I*ln(exp(b*x+a)-I*exp(a-d))/b*exp(2*a-2*d)+ln(exp(b *x+a)-exp(a-d))/b*exp(2*a-2*d)-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. 491 vs. \(2 (70) = 140\).
Time = 0.11 (sec) , antiderivative size = 491, normalized size of antiderivative = 6.46 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^2,x, algorithm="fricas")
Output:
(4*(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)*sin h(-2*a + 2*d))*sinh(b*x + d) - (cosh(b*x + d)^2 + 1)*sinh(-2*a + 2*d) + co sh(-2*a + 2*d))*arctan(cosh(b*x + d) + sinh(b*x + d)) - 2*cosh(b*x + d)*co sh(-2*a + 2*d) - (cosh(b*x + d)^2*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - s inh(-2*a + 2*d))*sinh(b*x + d)^2 + 2*(cosh(b*x + d)*cosh(-2*a + 2*d) - cos h(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) + (co sh(b*x + d)^2*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sin h(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) - 2*(cosh(-2*a + 2*d) - s inh(-2*a + 2*d))*sinh(b*x + d) + 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)
\[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=e^{2 a} \int e^{2 b x} \operatorname {csch}{\left (b x + d \right )} \operatorname {sech}^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)**2,x)
Output:
exp(2*a)*Integral(exp(2*b*x)*csch(b*x + d)*sech(b*x + d)**2, x)
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.36 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=-\frac {4 \, \arctan \left (e^{\left (-b x - d\right )}\right ) e^{\left (2 \, a - 2 \, d\right )}}{b} - \frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{b} + \frac {e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{b} - \frac {2 \, e^{\left (-b x + 2 \, a - 3 \, d\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, d\right )} + 1\right )}} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^2,x, algorithm="maxima")
Output:
-4*arctan(e^(-b*x - d))*e^(2*a - 2*d)/b - e^(2*a - 2*d)*log(e^(-b*x - d) + 1)/b + e^(2*a - 2*d)*log(e^(-b*x - d) - 1)/b - 2*e^(-b*x + 2*a - 3*d)/(b* (e^(-2*b*x - 2*d) + 1))
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\frac {{\left (4 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-2 \, d\right )} - e^{\left (-2 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) + e^{\left (-2 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) - \frac {2 \, e^{\left (b x - d\right )}}{e^{\left (2 \, b x + 2 \, d\right )} + 1}\right )} e^{\left (2 \, a\right )}}{b} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^2,x, algorithm="giac")
Output:
(4*arctan(e^(b*x + d))*e^(-2*d) - e^(-2*d)*log(e^(b*x + d) + 1) + e^(-2*d) *log(abs(e^(b*x + d) - 1)) - 2*e^(b*x - d)/(e^(2*b*x + 2*d) + 1))*e^(2*a)/ b
Time = 3.39 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.37 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\frac {{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (160\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}-160\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{b}-\frac {2\,{\mathrm {e}}^{2\,a-2\,d}\,\left (\ln \left (320\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}-{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,320{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (320\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}+{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,320{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{b}-\frac {{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (-160\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}-160\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{b}-\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b+b\,{\mathrm {e}}^{2\,d+2\,b\,x}} \] Input:
int(exp(2*a + 2*b*x)/(cosh(d + b*x)^2*sinh(d + b*x)),x)
Output:
(exp(2*a - 2*d)*log(160*exp(6*a)*exp(-6*d) - 160*exp(6*a)*exp(-5*d)*exp(b* x)))/b - (2*exp(2*a - 2*d)*(log(320*exp(6*a)*exp(-5*d)*exp(b*x) - exp(6*a) *exp(-6*d)*320i)*1i - log(exp(6*a)*exp(-6*d)*320i + 320*exp(6*a)*exp(-5*d) *exp(b*x))*1i))/b - (exp(2*a - 2*d)*log(- 160*exp(6*a)*exp(-6*d) - 160*exp (6*a)*exp(-5*d)*exp(b*x)))/b - (2*exp(2*a - d + b*x))/(b + b*exp(2*d + 2*b *x))
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.78 \[ \int e^{2 (a+b x)} \text {csch}(d+b x) \text {sech}^2(d+b x) \, dx=\frac {e^{2 a} \left (4 e^{2 b x +2 d} \mathit {atan} \left (e^{b x +d}\right )+4 \mathit {atan} \left (e^{b x +d}\right )+e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )-2 e^{b x +d}+\mathrm {log}\left (e^{b x +d}-1\right )-\mathrm {log}\left (e^{b x +d}+1\right )\right )}{e^{2 d} b \left (e^{2 b x +2 d}+1\right )} \] Input:
int(exp(2*b*x+2*a)*csch(b*x+d)*sech(b*x+d)^2,x)
Output:
(e**(2*a)*(4*e**(2*b*x + 2*d)*atan(e**(b*x + d)) + 4*atan(e**(b*x + 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))