Integrand size = 24, antiderivative size = 78 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=\frac {2 e^{2 a-d+b x}}{b \left (1-e^{2 d+2 b x}\right )}+\frac {2 e^{2 a-2 d} \arctan \left (e^{d+b x}\right )}{b}-\frac {4 e^{2 a-2 d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
2*exp(b*x+2*a-d)/b/(1-exp(2*b*x+2*d))+2*exp(2*a-2*d)*arctan(exp(b*x+d))/b- 4*exp(2*a-2*d)*arctanh(exp(b*x+d))/b
Time = 0.49 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=\frac {2 e^{2 a-2 d} \left (-\frac {e^{d+b x}}{-1+e^{2 (d+b x)}}+\arctan \left (e^{d+b x}\right )+\log \left (1-e^{d+b x}\right )-\log \left (1+e^{d+b x}\right )\right )}{b} \] Input:
Integrate[E^(2*(a + b*x))*Csch[d + b*x]^2*Sech[d + b*x],x]
Output:
(2*E^(2*a - 2*d)*(-(E^(d + b*x)/(-1 + E^(2*(d + b*x)))) + ArcTan[E^(d + b* x)] + Log[1 - E^(d + b*x)] - Log[1 + E^(d + b*x)]))/b
Time = 0.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65, 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.
\(\displaystyle \int e^{2 (a+b x)} \text {csch}^2(b x+d) \text {sech}(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {8 e^{2 a+4 b x}}{\left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {8 e^{2 a} \int \frac {e^{4 b x}}{\left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )}de^{b x}}{b}\) |
\(\Big \downarrow \) 372 |
\(\displaystyle \frac {8 e^{2 a} \left (\frac {e^{b x}}{4 \left (1-e^{2 b x}\right )}-\frac {1}{4} \int \frac {1+3 e^{2 b x}}{\left (1-e^{2 b x}\right ) \left (1+e^{2 b x}\right )}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {8 e^{2 a} \left (\frac {1}{4} \left (\int \frac {1}{1+e^{2 b x}}de^{b x}-2 \int \frac {1}{1-e^{2 b x}}de^{b x}\right )+\frac {e^{b x}}{4 \left (1-e^{2 b x}\right )}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {8 e^{2 a} \left (\frac {1}{4} \left (\arctan \left (e^{b x}\right )-2 \int \frac {1}{1-e^{2 b x}}de^{b x}\right )+\frac {e^{b x}}{4 \left (1-e^{2 b x}\right )}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {8 e^{2 a} \left (\frac {1}{4} \left (\arctan \left (e^{b x}\right )-2 \text {arctanh}\left (e^{b x}\right )\right )+\frac {e^{b x}}{4 \left (1-e^{2 b x}\right )}\right )}{b}\) |
Input:
Int[E^(2*(a + b*x))*Csch[d + b*x]^2*Sech[d + b*x],x]
Output:
(8*E^(2*a)*(E^(b*x)/(4*(1 - E^(2*b*x))) + (ArcTan[E^(b*x)] - 2*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 = 7.03 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.01
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +4 a -d}}{\left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right ) b}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}+\frac {2 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{2 a -2 d}}{b}\) | \(157\) |
Input:
int(exp(2*b*x+2*a)*csch(b*x+d)^2*sech(b*x+d),x,method=_RETURNVERBOSE)
Output:
2/(-exp(2*b*x+2*a+2*d)+exp(2*a))/b*exp(b*x+4*a-d)+I*ln(exp(b*x+a)+I*exp(a- d))/b*exp(2*a-2*d)-I*ln(exp(b*x+a)-I*exp(a-d))/b*exp(2*a-2*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. 498 vs. \(2 (70) = 140\).
Time = 0.10 (sec) , antiderivative size = 498, normalized size of antiderivative = 6.38 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx =\text {Too large to display} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)^2*sech(b*x+d),x, algorithm="fricas")
Output:
2*((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)) - cosh(b*x + d)*cosh (-2*a + 2*d) - (cosh(b*x + d)^2*cosh(-2*a + 2*d) + (cosh(-2*a + 2*d) - sin h(-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) - (cosh(-2*a + 2*d) - sinh( -2*a + 2*d))*sinh(b*x + d) + 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)
\[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=e^{2 a} \int e^{2 b x} \operatorname {csch}^{2}{\left (b x + d \right )} \operatorname {sech}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)**2*sech(b*x+d),x)
Output:
exp(2*a)*Integral(exp(2*b*x)*csch(b*x + d)**2*sech(b*x + d), x)
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.33 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=-\frac {2 \, \arctan \left (e^{\left (-b x - d\right )}\right ) e^{\left (2 \, a - 2 \, d\right )}}{b} - \frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} + 1\right )}{b} + \frac {2 \, e^{\left (2 \, a - 2 \, d\right )} \log \left (e^{\left (-b x - d\right )} - 1\right )}{b} + \frac {2 \, e^{\left (-b x + 2 \, a - 3 \, d\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, d\right )} - 1\right )}} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)^2*sech(b*x+d),x, algorithm="maxima")
Output:
-2*arctan(e^(-b*x - d))*e^(2*a - 2*d)/b - 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*e^(-b*x + 2*a - 3*d) /(b*(e^(-2*b*x - 2*d) - 1))
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=\frac {2 \, {\left (\arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-2 \, d\right )} - e^{\left (-2 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right ) + e^{\left (-2 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right ) - \frac {e^{\left (b x - d\right )}}{e^{\left (2 \, b x + 2 \, d\right )} - 1}\right )} e^{\left (2 \, a\right )}}{b} \] Input:
integrate(exp(2*b*x+2*a)*csch(b*x+d)^2*sech(b*x+d),x, algorithm="giac")
Output:
2*(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)) - e^(b*x - d)/(e^(2*b*x + 2*d) - 1))*e^(2*a)/b
Time = 4.18 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.33 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=\frac {2\,{\mathrm {e}}^{2\,a-d+b\,x}}{b-b\,{\mathrm {e}}^{2\,d+2\,b\,x}}-\frac {{\mathrm {e}}^{2\,a-2\,d}\,\left (\ln \left (160\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}-{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,160{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (160\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}+{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}\,160{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{b}-\frac {2\,{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (-320\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}-320\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{b}+\frac {2\,{\mathrm {e}}^{2\,a-2\,d}\,\ln \left (320\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-6\,d}-320\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{-5\,d}\,{\mathrm {e}}^{b\,x}\right )}{b} \] Input:
int(exp(2*a + 2*b*x)/(cosh(d + b*x)*sinh(d + b*x)^2),x)
Output:
(2*exp(2*a - d + b*x))/(b - b*exp(2*d + 2*b*x)) - (exp(2*a - 2*d)*(log(160 *exp(6*a)*exp(-5*d)*exp(b*x) - exp(6*a)*exp(-6*d)*160i)*1i - log(exp(6*a)* exp(-6*d)*160i + 160*exp(6*a)*exp(-5*d)*exp(b*x))*1i))/b - (2*exp(2*a - 2* d)*log(- 320*exp(6*a)*exp(-6*d) - 320*exp(6*a)*exp(-5*d)*exp(b*x)))/b + (2 *exp(2*a - 2*d)*log(320*exp(6*a)*exp(-6*d) - 320*exp(6*a)*exp(-5*d)*exp(b* x)))/b
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.73 \[ \int e^{2 (a+b x)} \text {csch}^2(d+b x) \text {sech}(d+b x) \, dx=\frac {2 e^{2 a} \left (e^{2 b x +2 d} \mathit {atan} \left (e^{b x +d}\right )-\mathit {atan} \left (e^{b x +d}\right )+e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}-1\right )-e^{2 b x +2 d} \mathrm {log}\left (e^{b x +d}+1\right )-e^{b x +d}-\mathrm {log}\left (e^{b x +d}-1\right )+\mathrm {log}\left (e^{b x +d}+1\right )\right )}{e^{2 d} b \left (e^{2 b x +2 d}-1\right )} \] Input:
int(exp(2*b*x+2*a)*csch(b*x+d)^2*sech(b*x+d),x)
Output:
(2*e**(2*a)*(e**(2*b*x + 2*d)*atan(e**(b*x + d)) - 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) - 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))