Integrand size = 20, antiderivative size = 279 \[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=\frac {3 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{5 b}-\frac {2 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{b \left (1-e^{2 (d+b x)}\right )^2}+\frac {11 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{3 b \left (1-e^{2 (d+b x)}\right )}-\frac {43 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1-2 e^{\frac {1}{3} (d+b x)}}{\sqrt {3}}\right )}{6 \sqrt {3} b}+\frac {43 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1+2 e^{\frac {1}{3} (d+b x)}}{\sqrt {3}}\right )}{6 \sqrt {3} b}-\frac {43 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (e^{\frac {1}{3} (d+b x)}\right )}{9 b}-\frac {43 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (\frac {e^{\frac {1}{3} (d+b x)}}{1+e^{\frac {2}{3} (d+b x)}}\right )}{18 b} \] Output:
3/5*exp(5/3*b*x+5/3*a)/b-2*exp(5/3*b*x+5/3*a)/b/(1-exp(2*b*x+2*d))^2+11/3* exp(5/3*b*x+5/3*a)/b/(1-exp(2*b*x+2*d))-43/18*3^(1/2)*exp(5/3*a-5/3*d)*arc tan(1/3*(1-2*exp(1/3*b*x+1/3*d))*3^(1/2))/b+43/18*3^(1/2)*exp(5/3*a-5/3*d) *arctan(1/3*(1+2*exp(1/3*b*x+1/3*d))*3^(1/2))/b-43/9*exp(5/3*a-5/3*d)*arct anh(exp(1/3*b*x+1/3*d))/b-43/18*exp(5/3*a-5/3*d)*arctanh(exp(1/3*b*x+1/3*d )/(1+exp(2/3*b*x+2/3*d)))/b
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.61 \[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=\frac {e^{5 a/3} \left (162 e^{\frac {5 b x}{3}}-215 \text {RootSum}\left [-\cosh (d)+\sinh (d)+\cosh (d) \text {$\#$1}^6+\sinh (d) \text {$\#$1}^6\&,\frac {b x-3 \log \left (e^{\frac {b x}{3}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] (\cosh (d)-\sinh (d))^2-\frac {540 e^{\frac {5 b x}{3}} (\cosh (d)-\sinh (d))^2}{\left (\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)\right )^2}-\frac {990 e^{\frac {5 b x}{3}} (\cosh (d)-\sinh (d))}{\left (-1+e^{2 b x}\right ) \cosh (d)+\left (1+e^{2 b x}\right ) \sinh (d)}\right )}{270 b} \] Input:
Integrate[E^((5*(a + b*x))/3)*Coth[d + b*x]^3,x]
Output:
(E^((5*a)/3)*(162*E^((5*b*x)/3) - 215*RootSum[-Cosh[d] + Sinh[d] + Cosh[d] *#1^6 + Sinh[d]*#1^6 & , (b*x - 3*Log[E^((b*x)/3) - #1])/#1 & ]*(Cosh[d] - Sinh[d])^2 - (540*E^((5*b*x)/3)*(Cosh[d] - Sinh[d])^2)/((-1 + E^(2*b*x))* Cosh[d] + (1 + E^(2*b*x))*Sinh[d])^2 - (990*E^((5*b*x)/3)*(Cosh[d] - Sinh[ d]))/((-1 + E^(2*b*x))*Cosh[d] + (1 + E^(2*b*x))*Sinh[d])))/(270*b)
Time = 0.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.82, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {2720, 25, 27, 968, 27, 1047, 27, 959, 825, 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.
| \(\displaystyle \int e^{\frac {5}{3} (a+b x)} \coth ^3(b x+d) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \int -\frac {e^{\frac {5 a}{3}+\frac {4 b x}{3}} \left (1+e^{2 b x}\right )^3}{\left (1-e^{2 b x}\right )^3}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \int \frac {e^{\frac {5 a}{3}+\frac {4 b x}{3}} \left (1+e^{2 b x}\right )^3}{\left (1-e^{2 b x}\right )^3}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \int \frac {e^{\frac {4 b x}{3}} \left (1+e^{2 b x}\right )^3}{\left (1-e^{2 b x}\right )^3}de^{\frac {b x}{3}}}{b}\) |
| \(\Big \downarrow \) 968 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{12} \int \frac {2 e^{\frac {4 b x}{3}} \left (1-11 e^{2 b x}\right ) \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2}de^{\frac {b x}{3}}+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \int \frac {e^{\frac {4 b x}{3}} \left (1-11 e^{2 b x}\right ) \left (1+e^{2 b x}\right )}{\left (1-e^{2 b x}\right )^2}de^{\frac {b x}{3}}+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1047 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {1}{6} \int -\frac {4 e^{\frac {4 b x}{3}} \left (1-44 e^{2 b x}\right )}{1-e^{2 b x}}de^{\frac {b x}{3}}+\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \int \frac {e^{\frac {4 b x}{3}} \left (1-44 e^{2 b x}\right )}{1-e^{2 b x}}de^{\frac {b x}{3}}\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \int \frac {e^{\frac {4 b x}{3}}}{1-e^{2 b x}}de^{\frac {b x}{3}}\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{3} \int \frac {1}{1-e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{3} \int -\frac {1+e^{\frac {b x}{3}}}{2 \left (1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}\right )}de^{\frac {b x}{3}}+\frac {1}{3} \int -\frac {1-e^{\frac {b x}{3}}}{2 \left (1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}\right )}de^{\frac {b x}{3}}\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{3} \int \frac {1}{1-e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1+e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1-e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (-\frac {1}{6} \int \frac {1+e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1-e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{2} \int -\frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {3}{2} \int \frac {1}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {3}{2} \int \frac {1}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {3}{2} \int \frac {1}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (3 \int \frac {1}{-3-e^{\frac {2 b x}{3}}}d\left (-1+2 e^{\frac {b x}{3}}\right )+\frac {1}{2} \int \frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-e^{\frac {2 b x}{3}}}d\left (1+2 e^{\frac {b x}{3}}\right )+\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 e^{\frac {b x}{3}}}{1-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1+2 e^{\frac {b x}{3}}}{1+e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (1-11 e^{2 b x}\right )}{3 \left (1-e^{2 b x}\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (-e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}+1\right )\right )+\frac {1}{6} \left (\frac {1}{2} \log \left (e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}+1\right )-\sqrt {3} \arctan \left (\frac {2 e^{\frac {b x}{3}}+1}{\sqrt {3}}\right )\right )+\frac {1}{3} \text {arctanh}\left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (e^{2 b x}+1\right )^2}{6 \left (1-e^{2 b x}\right )^2}\right )}{b}\) |
Input:
Int[E^((5*(a + b*x))/3)*Coth[d + b*x]^3,x]
Output:
(-3*E^((5*a)/3)*((E^((5*b*x)/3)*(1 + E^(2*b*x))^2)/(6*(1 - E^(2*b*x))^2) + ((E^((5*b*x)/3)*(1 - 11*E^(2*b*x)))/(3*(1 - E^(2*b*x))) - (2*((44*E^((5*b *x)/3))/5 - 43*(ArcTanh[E^((b*x)/3)]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*E^((b*x )/3))/Sqrt[3]]) - Log[1 - E^((b*x)/3) + E^((2*b*x)/3)]/2)/6 + (-(Sqrt[3]*A rcTan[(1 + 2*E^((b*x)/3))/Sqrt[3]]) + Log[1 + E^((b*x)/3) + E^((2*b*x)/3)] /2)/6)))/3)/6))/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[(-(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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) *((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1)) Int [(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c *b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^( m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Simp[1/( a*b*n*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c *(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IGtQ[n , 0] && LtQ[p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
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 = 0.31 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(\frac {3 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{5 b}-\frac {\left (11 \,{\mathrm e}^{2 b x +2 d}-5\right ) {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{3 \left ({\mathrm e}^{2 b x +2 d}-1\right )^{2} b}+\frac {43 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{36 b}+\frac {43 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{36 b}+\frac {43 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{36 b}-\frac {43 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{36 b}+\frac {43 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-1\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{18 b}-\frac {43 \ln \left (1+{\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{18 b}-\frac {43 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{36 b}+\frac {43 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{36 b}-\frac {43 \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{36 b}-\frac {43 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}} \sqrt {3}}{36 b}\) | \(370\) |
Input:
int(exp(5/3*b*x+5/3*a)*coth(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
3/5*exp(5/3*b*x+5/3*a)/b-1/3/(exp(2*b*x+2*d)-1)^2/b*(11*exp(2*b*x+2*d)-5)* exp(5/3*b*x+5/3*a)+43/36/b*ln(exp(1/3*b*x+1/3*d)-1/2+1/2*I*3^(1/2))*exp(5/ 3*a-5/3*d)+43/36*I/b*ln(exp(1/3*b*x+1/3*d)-1/2+1/2*I*3^(1/2))*exp(5/3*a-5/ 3*d)*3^(1/2)+43/36/b*ln(exp(1/3*b*x+1/3*d)-1/2-1/2*I*3^(1/2))*exp(5/3*a-5/ 3*d)-43/36*I/b*ln(exp(1/3*b*x+1/3*d)-1/2-1/2*I*3^(1/2))*exp(5/3*a-5/3*d)*3 ^(1/2)+43/18/b*ln(exp(1/3*b*x+1/3*d)-1)*exp(5/3*a-5/3*d)-43/18/b*ln(1+exp( 1/3*b*x+1/3*d))*exp(5/3*a-5/3*d)-43/36/b*ln(exp(1/3*b*x+1/3*d)+1/2+1/2*I*3 ^(1/2))*exp(5/3*a-5/3*d)+43/36*I/b*ln(exp(1/3*b*x+1/3*d)+1/2+1/2*I*3^(1/2) )*exp(5/3*a-5/3*d)*3^(1/2)-43/36/b*ln(exp(1/3*b*x+1/3*d)+1/2-1/2*I*3^(1/2) )*exp(5/3*a-5/3*d)-43/36*I/b*ln(exp(1/3*b*x+1/3*d)+1/2-1/2*I*3^(1/2))*exp( 5/3*a-5/3*d)*3^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 7903 vs. \(2 (199) = 398\).
Time = 0.19 (sec) , antiderivative size = 7903, normalized size of antiderivative = 28.33 \[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)*coth(b*x+d)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=e^{\frac {5 a}{3}} \int e^{\frac {5 b x}{3}} \coth ^{3}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(5/3*b*x+5/3*a)*coth(b*x+d)**3,x)
Output:
exp(5*a/3)*Integral(exp(5*b*x/3)*coth(b*x + d)**3, x)
Time = 0.12 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.93 \[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=-\frac {43 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + 1\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{18 \, b} - \frac {43 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - 1\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{18 \, b} - \frac {43 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right )}{36 \, b} - \frac {43 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + 1\right )}{18 \, b} + \frac {43 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - 1\right )}{18 \, b} + \frac {43 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (-e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right )}{36 \, b} - \frac {{\left (73 \, e^{\left (-2 \, b x - 2 \, d\right )} - 34 \, e^{\left (-4 \, b x - 4 \, d\right )} - 9\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{15 \, b {\left (e^{\left (-\frac {5}{3} \, b x - \frac {5}{3} \, d\right )} - 2 \, e^{\left (-\frac {11}{3} \, b x - \frac {11}{3} \, d\right )} + e^{\left (-\frac {17}{3} \, b x - \frac {17}{3} \, d\right )}\right )}} \] Input:
integrate(exp(5/3*b*x+5/3*a)*coth(b*x+d)^3,x, algorithm="maxima")
Output:
-43/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(-1/3*b*x - 1/3*d) + 1))*e^(5/3*a - 5/3*d)/b - 43/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(-1/3*b*x - 1/3*d) - 1)) *e^(5/3*a - 5/3*d)/b - 43/36*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/3*d) + e^(-2/3*b*x - 2/3*d) + 1)/b - 43/18*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/ 3*d) + 1)/b + 43/18*e^(5/3*a - 5/3*d)*log(e^(-1/3*b*x - 1/3*d) - 1)/b + 43 /36*e^(5/3*a - 5/3*d)*log(-e^(-1/3*b*x - 1/3*d) + e^(-2/3*b*x - 2/3*d) + 1 )/b - 1/15*(73*e^(-2*b*x - 2*d) - 34*e^(-4*b*x - 4*d) - 9)*e^(5/3*a - 5/3* d)/(b*(e^(-5/3*b*x - 5/3*d) - 2*e^(-11/3*b*x - 11/3*d) + e^(-17/3*b*x - 17 /3*d)))
Time = 0.13 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.85 \[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=\frac {430 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (\frac {1}{3} \, b x\right )} + e^{\left (-\frac {1}{3} \, d\right )}\right )} e^{\left (\frac {1}{3} \, d\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} + 430 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, e^{\left (\frac {1}{3} \, b x\right )} - e^{\left (-\frac {1}{3} \, d\right )}\right )} e^{\left (\frac {1}{3} \, d\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} - 215 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right ) + 215 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (-e^{\left (\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right ) - 430 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (e^{\left (\frac {1}{3} \, b x\right )} + e^{\left (-\frac {1}{3} \, d\right )}\right ) + 430 \, e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left ({\left | e^{\left (\frac {1}{3} \, b x\right )} - e^{\left (-\frac {1}{3} \, d\right )} \right |}\right ) - \frac {60 \, {\left (11 \, e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 2 \, d\right )} - 5 \, e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} - 1\right )}^{2}} + 108 \, e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a\right )}}{180 \, b} \] Input:
integrate(exp(5/3*b*x+5/3*a)*coth(b*x+d)^3,x, algorithm="giac")
Output:
1/180*(430*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(1/3*b*x) + e^(-1/3*d))*e^(1/3* d))*e^(5/3*a - 5/3*d) + 430*sqrt(3)*arctan(1/3*sqrt(3)*(2*e^(1/3*b*x) - e^ (-1/3*d))*e^(1/3*d))*e^(5/3*a - 5/3*d) - 215*e^(5/3*a - 5/3*d)*log(e^(1/3* b*x - 1/3*d) + e^(2/3*b*x) + e^(-2/3*d)) + 215*e^(5/3*a - 5/3*d)*log(-e^(1 /3*b*x - 1/3*d) + e^(2/3*b*x) + e^(-2/3*d)) - 430*e^(5/3*a - 5/3*d)*log(e^ (1/3*b*x) + e^(-1/3*d)) + 430*e^(5/3*a - 5/3*d)*log(abs(e^(1/3*b*x) - e^(- 1/3*d))) - 60*(11*e^(11/3*b*x + 5/3*a + 2*d) - 5*e^(5/3*b*x + 5/3*a))/(e^( 2*b*x + 2*d) - 1)^2 + 108*e^(5/3*b*x + 5/3*a))/b
Time = 5.43 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.74 \[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx =\text {Too large to display} \] Input:
int(coth(d + b*x)^3*exp((5*a)/3 + (5*b*x)/3),x)
Output:
(3*exp((5*a)/3 + (5*b*x)/3))/(5*b) - (11*exp((5*a)/3 + (5*b*x)/3))/(3*b*(e xp(2*d + 2*b*x) - 1)) - (2*exp((5*a)/3 + (5*b*x)/3))/(b*(exp(4*d + 4*b*x) - 2*exp(2*d + 2*b*x) + 1)) - (43*exp(10*a - 10*d)^(1/6)*log(- (1849*exp((1 0*a)/3)*exp(-(10*d)/3))/81 - (1849*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp ((b*x)/3)*(exp(10*a)*exp(-10*d))^(1/6))/81))/(18*b) + (43*exp(10*a - 10*d) ^(1/6)*log((1849*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*(exp(10* a)*exp(-10*d))^(1/6))/81 - (1849*exp((10*a)/3)*exp(-(10*d)/3))/81))/(18*b) - (43*log(- (1849*exp((10*a)/3)*exp(-(10*d)/3))/81 - (1849*exp((5*a)/3)*e xp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1i)/2 - 1/2)*(exp(10*a)*exp(- 10*d))^(1/6))/81)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 - 1/2))/(18*b) + (43*log((1849*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1 i)/2 - 1/2)*(exp(10*a)*exp(-10*d))^(1/6))/81 - (1849*exp((10*a)/3)*exp(-(1 0*d)/3))/81)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 - 1/2))/(18*b) - (43*l og(- (1849*exp((10*a)/3)*exp(-(10*d)/3))/81 - (1849*exp((5*a)/3)*exp(d/3)* exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1i)/2 + 1/2)*(exp(10*a)*exp(-10*d))^( 1/6))/81)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 + 1/2))/(18*b) + (43*log( (1849*exp((5*a)/3)*exp(d/3)*exp(-(5*d)/3)*exp((b*x)/3)*((3^(1/2)*1i)/2 + 1 /2)*(exp(10*a)*exp(-10*d))^(1/6))/81 - (1849*exp((10*a)/3)*exp(-(10*d)/3)) /81)*exp(10*a - 10*d)^(1/6)*((3^(1/2)*1i)/2 + 1/2))/(18*b)
\[ \int e^{\frac {5}{3} (a+b x)} \coth ^3(d+b x) \, dx=\int e^{\frac {5 b x}{3}+\frac {5 a}{3}} \coth \left (b x +d \right )^{3}d x \] Input:
int(exp(5/3*b*x+5/3*a)*coth(b*x+d)^3,x)
Output:
int(e**((5*a + 5*b*x)/3)*coth(b*x + d)**3,x)