Integrand size = 10, antiderivative size = 105 \[ \int e^x \text {csch}^2(3 x) \, dx=\frac {2 e^x}{3 \left (1-e^{6 x}\right )}+\frac {\arctan \left (\frac {1-2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+2 e^x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2 \text {arctanh}\left (e^x\right )}{9}+\frac {1}{18} \log \left (1-e^x+e^{2 x}\right )-\frac {1}{18} \log \left (1+e^x+e^{2 x}\right ) \]
2/3*exp(x)/(1-exp(6*x))-2/9*arctanh(exp(x))+1/18*ln(1-exp(x)+exp(2*x))-1/1 8*ln(1+exp(x)+exp(2*x))+1/9*arctan(1/3*(1-2*exp(x))*3^(1/2))*3^(1/2)-1/9*a rctan(1/3*(1+2*exp(x))*3^(1/2))*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int e^x \text {csch}^2(3 x) \, dx=\frac {2}{3} e^x \left (\frac {1}{1-e^{6 x}}-\operatorname {Hypergeometric2F1}\left (\frac {1}{6},1,\frac {7}{6},e^{6 x}\right )\right ) \]
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {2720, 27, 817, 754, 27, 219, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int e^x \text {csch}^2(3 x) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int \frac {4 e^{6 x}}{\left (1-e^{6 x}\right )^2}de^x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {e^{6 x}}{\left (1-e^{6 x}\right )^2}de^x\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 4 \left (\frac {e^x}{6 \left (1-e^{6 x}\right )}-\frac {1}{6} \int \frac {1}{1-e^{6 x}}de^x\right )\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (-\frac {1}{3} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{3} \int \frac {2-e^x}{2 \left (1-e^x+e^{2 x}\right )}de^x-\frac {1}{3} \int \frac {2+e^x}{2 \left (1+e^x+e^{2 x}\right )}de^x\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (-\frac {1}{3} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{6} \int \frac {2-e^x}{1-e^x+e^{2 x}}de^x-\frac {1}{6} \int \frac {2+e^x}{1+e^x+e^{2 x}}de^x\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (-\frac {1}{6} \int \frac {2-e^x}{1-e^x+e^{2 x}}de^x-\frac {1}{6} \int \frac {2+e^x}{1+e^x+e^{2 x}}de^x-\frac {1}{3} \text {arctanh}\left (e^x\right )\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (\frac {1}{6} \left (\frac {1}{2} \int -\frac {1-2 e^x}{1-e^x+e^{2 x}}de^x-\frac {3}{2} \int \frac {1}{1-e^x+e^{2 x}}de^x\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{1+e^x+e^{2 x}}de^x-\frac {1}{2} \int \frac {1+2 e^x}{1+e^x+e^{2 x}}de^x\right )-\frac {1}{3} \text {arctanh}\left (e^x\right )\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{1-e^x+e^{2 x}}de^x-\frac {1}{2} \int \frac {1-2 e^x}{1-e^x+e^{2 x}}de^x\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{1+e^x+e^{2 x}}de^x-\frac {1}{2} \int \frac {1+2 e^x}{1+e^x+e^{2 x}}de^x\right )-\frac {1}{3} \text {arctanh}\left (e^x\right )\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (\frac {1}{6} \left (3 \int \frac {1}{-3-e^{2 x}}d\left (-1+2 e^x\right )-\frac {1}{2} \int \frac {1-2 e^x}{1-e^x+e^{2 x}}de^x\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-e^{2 x}}d\left (1+2 e^x\right )-\frac {1}{2} \int \frac {1+2 e^x}{1+e^x+e^{2 x}}de^x\right )-\frac {1}{3} \text {arctanh}\left (e^x\right )\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (\frac {1}{6} \left (-\frac {1}{2} \int \frac {1-2 e^x}{1-e^x+e^{2 x}}de^x-\sqrt {3} \arctan \left (\frac {2 e^x-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\frac {1}{2} \int \frac {1+2 e^x}{1+e^x+e^{2 x}}de^x-\sqrt {3} \arctan \left (\frac {2 e^x+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \text {arctanh}\left (e^x\right )\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 4 \left (\frac {1}{6} \left (\frac {1}{6} \left (\frac {1}{2} \log \left (-e^x+e^{2 x}+1\right )-\sqrt {3} \arctan \left (\frac {2 e^x-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 e^x+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (e^x+e^{2 x}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (e^x\right )\right )+\frac {e^x}{6 \left (1-e^{6 x}\right )}\right )\) |
4*(E^x/(6*(1 - E^(6*x))) + (-1/3*ArcTanh[E^x] + (-(Sqrt[3]*ArcTan[(-1 + 2* E^x)/Sqrt[3]]) + Log[1 - E^x + E^(2*x)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + 2*E^ x)/Sqrt[3]]) - Log[1 + E^x + E^(2*x)]/2)/6)/6)
3.4.17.3.1 Defintions of rubi rules used
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, 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 = 0.74 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(-\frac {2 \,{\mathrm e}^{x}}{3 \left ({\mathrm e}^{6 x}-1\right )}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{x}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {\ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{x}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{9}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{9}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}+\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{18}-\frac {\ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}-\frac {i \sqrt {3}\, \ln \left ({\mathrm e}^{x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{18}\) | \(148\) |
-2/3*exp(x)/(exp(6*x)-1)+1/18*ln(exp(x)-1/2-1/2*I*3^(1/2))+1/18*I*3^(1/2)* ln(exp(x)-1/2-1/2*I*3^(1/2))+1/18*ln(exp(x)-1/2+1/2*I*3^(1/2))-1/18*I*3^(1 /2)*ln(exp(x)-1/2+1/2*I*3^(1/2))+1/9*ln(exp(x)-1)-1/9*ln(exp(x)+1)-1/18*ln (exp(x)+1/2-1/2*I*3^(1/2))+1/18*I*3^(1/2)*ln(exp(x)+1/2-1/2*I*3^(1/2))-1/1 8*ln(exp(x)+1/2+1/2*I*3^(1/2))-1/18*I*3^(1/2)*ln(exp(x)+1/2+1/2*I*3^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (76) = 152\).
Time = 0.28 (sec) , antiderivative size = 560, normalized size of antiderivative = 5.33 \[ \int e^x \text {csch}^2(3 x) \, dx=\text {Too large to display} \]
-1/18*(2*(sqrt(3)*cosh(x)^6 + 6*sqrt(3)*cosh(x)^5*sinh(x) + 15*sqrt(3)*cos h(x)^4*sinh(x)^2 + 20*sqrt(3)*cosh(x)^3*sinh(x)^3 + 15*sqrt(3)*cosh(x)^2*s inh(x)^4 + 6*sqrt(3)*cosh(x)*sinh(x)^5 + sqrt(3)*sinh(x)^6 - sqrt(3))*arct an(2/3*sqrt(3)*cosh(x) + 2/3*sqrt(3)*sinh(x) + 1/3*sqrt(3)) + 2*(sqrt(3)*c osh(x)^6 + 6*sqrt(3)*cosh(x)^5*sinh(x) + 15*sqrt(3)*cosh(x)^4*sinh(x)^2 + 20*sqrt(3)*cosh(x)^3*sinh(x)^3 + 15*sqrt(3)*cosh(x)^2*sinh(x)^4 + 6*sqrt(3 )*cosh(x)*sinh(x)^5 + sqrt(3)*sinh(x)^6 - sqrt(3))*arctan(2/3*sqrt(3)*cosh (x) + 2/3*sqrt(3)*sinh(x) - 1/3*sqrt(3)) + (cosh(x)^6 + 6*cosh(x)^5*sinh(x ) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x) ^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 - 1)*log((2*cosh(x) + 1)/(cosh(x) - s inh(x))) - (cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20* cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh( x)^6 - 1)*log((2*cosh(x) - 1)/(cosh(x) - sinh(x))) + 2*(cosh(x)^6 + 6*cosh (x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh( x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 - 1)*log(cosh(x) + sinh(x ) + 1) - 2*(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20* cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh( x)^6 - 1)*log(cosh(x) + sinh(x) - 1) + 12*cosh(x) + 12*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 - 1)
\[ \int e^x \text {csch}^2(3 x) \, dx=\int e^{x} \operatorname {csch}^{2}{\left (3 x \right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.81 \[ \int e^x \text {csch}^2(3 x) \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {2 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 1\right )}} - \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{9} \, \log \left (e^{x} + 1\right ) + \frac {1}{9} \, \log \left (e^{x} - 1\right ) \]
-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x + 1)) - 1/9*sqrt(3)*arctan(1/3*sqrt (3)*(2*e^x - 1)) - 2/3*e^x/(e^(6*x) - 1) - 1/18*log(e^(2*x) + e^x + 1) + 1 /18*log(e^(2*x) - e^x + 1) - 1/9*log(e^x + 1) + 1/9*log(e^x - 1)
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int e^x \text {csch}^2(3 x) \, dx=-\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} + 1\right )}\right ) - \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{x} - 1\right )}\right ) - \frac {2 \, e^{x}}{3 \, {\left (e^{\left (6 \, x\right )} - 1\right )}} - \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} + e^{x} + 1\right ) + \frac {1}{18} \, \log \left (e^{\left (2 \, x\right )} - e^{x} + 1\right ) - \frac {1}{9} \, \log \left (e^{x} + 1\right ) + \frac {1}{9} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^x + 1)) - 1/9*sqrt(3)*arctan(1/3*sqrt (3)*(2*e^x - 1)) - 2/3*e^x/(e^(6*x) - 1) - 1/18*log(e^(2*x) + e^x + 1) + 1 /18*log(e^(2*x) - e^x + 1) - 1/9*log(e^x + 1) + 1/9*log(abs(e^x - 1))
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87 \[ \int e^x \text {csch}^2(3 x) \, dx=\frac {\ln \left (\frac {2}{3}-\frac {2\,{\mathrm {e}}^x}{3}\right )}{9}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x}{3}-\frac {2}{3}\right )}{9}+\frac {\ln \left ({\left (\frac {2\,{\mathrm {e}}^x}{3}-\frac {1}{3}\right )}^2+\frac {1}{3}\right )}{18}-\frac {\ln \left ({\left (\frac {2\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )}^2+\frac {1}{3}\right )}{18}-\frac {2\,{\mathrm {e}}^x}{3\,\left ({\mathrm {e}}^{6\,x}-1\right )}-\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,{\mathrm {e}}^x}{3}-\frac {1}{3}\right )\right )}{9}-\frac {\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\left (\frac {2\,{\mathrm {e}}^x}{3}+\frac {1}{3}\right )\right )}{9} \]