Integrand size = 20, antiderivative size = 271 \[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=\frac {3 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{5 b}-\frac {2 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{b \left (1+e^{2 (d+b x)}\right )^2}+\frac {11 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{3 b \left (1+e^{2 (d+b x)}\right )}-\frac {43 e^{\frac {5 (a-d)}{3}} \arctan \left (e^{\frac {1}{3} (d+b x)}\right )}{9 b}+\frac {43 e^{\frac {5 (a-d)}{3}} \arctan \left (\sqrt {3}-2 e^{\frac {1}{3} (d+b x)}\right )}{18 b}-\frac {43 e^{\frac {5 (a-d)}{3}} \arctan \left (\sqrt {3}+2 e^{\frac {1}{3} (d+b x)}\right )}{18 b}+\frac {43 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (\frac {\sqrt {3} e^{\frac {1}{3} (d+b x)}}{1+e^{\frac {2}{3} (d+b x)}}\right )}{6 \sqrt {3} b} \] 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))-43/9*exp(5/3*a-5/3*d)*arctan(exp(1 /3*b*x+1/3*d))/b-43/18*exp(5/3*a-5/3*d)*arctan(-3^(1/2)+2*exp(1/3*b*x+1/3* d))/b-43/18*exp(5/3*a-5/3*d)*arctan(3^(1/2)+2*exp(1/3*b*x+1/3*d))/b+43/18* 3^(1/2)*exp(5/3*a-5/3*d)*arctanh(3^(1/2)*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 = 0.63 \[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=\frac {e^{5 a/3} \left (162 e^{\frac {5 b x}{3}}+215 \text {RootSum}\left [\cosh (d)-\sinh (d)+\cosh (d) \text {$\#$1}^6+\sinh (d) \text {$\#$1}^6\&,\frac {b x-3 \log \left (e^{\frac {b x}{3}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] (\cosh (d)-\sinh (d))^2-\frac {540 e^{\frac {5 b x}{3}} (\cosh (d)-\sinh (d))^2}{\left (\left (1+e^{2 b x}\right ) \cosh (d)+\left (-1+e^{2 b x}\right ) \sinh (d)\right )^2}+\frac {990 e^{\frac {5 b x}{3}} (\cosh (d)-\sinh (d))}{\left (1+e^{2 b x}\right ) \cosh (d)+\left (-1+e^{2 b x}\right ) \sinh (d)}\right )}{270 b} \] Input:
Integrate[E^((5*(a + b*x))/3)*Tanh[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))*Co sh[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 = 0.43 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.86, 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, 824, 27, 216, 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.
| \(\displaystyle \int e^{\frac {5}{3} (a+b x)} \tanh ^3(b x+d) \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}-\frac {1}{12} \int -\frac {2 e^{\frac {4 b x}{3}} \left (1-e^{2 b x}\right ) \left (1+11 e^{2 b x}\right )}{\left (1+e^{2 b x}\right )^2}de^{\frac {b x}{3}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \int \frac {e^{\frac {4 b x}{3}} \left (1-e^{2 b x}\right ) \left (1+11 e^{2 b x}\right )}{\left (1+e^{2 b x}\right )^2}de^{\frac {b x}{3}}+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1047 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {1}{6} \int \frac {4 e^{\frac {4 b x}{3}} \left (1+44 e^{2 b x}\right )}{1+e^{2 b x}}de^{\frac {b x}{3}}\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \int \frac {e^{\frac {4 b x}{3}} \left (1+44 e^{2 b x}\right )}{1+e^{2 b x}}de^{\frac {b x}{3}}\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \int \frac {e^{\frac {4 b x}{3}}}{1+e^{2 b x}}de^{\frac {b x}{3}}\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{3} \int \frac {1}{1+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{3} \int -\frac {1-\sqrt {3} e^{\frac {b x}{3}}}{2 \left (1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}\right )}de^{\frac {b x}{3}}+\frac {1}{3} \int -\frac {1+\sqrt {3} e^{\frac {b x}{3}}}{2 \left (1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}\right )}de^{\frac {b x}{3}}\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{3} \int \frac {1}{1+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1-\sqrt {3} e^{\frac {b x}{3}}}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1+\sqrt {3} e^{\frac {b x}{3}}}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (-\frac {1}{6} \int \frac {1-\sqrt {3} e^{\frac {b x}{3}}}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{6} \int \frac {1+\sqrt {3} e^{\frac {b x}{3}}}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{3} \arctan \left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 e^{\frac {b x}{3}}}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}+2 e^{\frac {b x}{3}}}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \arctan \left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 e^{\frac {b x}{3}}}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}+2 e^{\frac {b x}{3}}}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \arctan \left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (-\int \frac {1}{-1-e^{\frac {2 b x}{3}}}d\left (-\sqrt {3}+2 e^{\frac {b x}{3}}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 e^{\frac {b x}{3}}}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{6} \left (-\int \frac {1}{-1-e^{\frac {2 b x}{3}}}d\left (\sqrt {3}+2 e^{\frac {b x}{3}}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}+2 e^{\frac {b x}{3}}}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \arctan \left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 e^{\frac {b x}{3}}}{1-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}-\arctan \left (\sqrt {3}-2 e^{\frac {b x}{3}}\right )\right )+\frac {1}{6} \left (\arctan \left (2 e^{\frac {b x}{3}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}+2 e^{\frac {b x}{3}}}{1+\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}}de^{\frac {b x}{3}}\right )+\frac {1}{3} \arctan \left (e^{\frac {b x}{3}}\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {3 e^{5 a/3} \left (\frac {1}{6} \left (\frac {e^{\frac {5 b x}{3}} \left (11 e^{2 b x}+1\right )}{3 \left (e^{2 b x}+1\right )}-\frac {2}{3} \left (\frac {44}{5} e^{\frac {5 b x}{3}}-43 \left (\frac {1}{3} \arctan \left (e^{\frac {b x}{3}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (-\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}+1\right )-\arctan \left (\sqrt {3}-2 e^{\frac {b x}{3}}\right )\right )+\frac {1}{6} \left (\arctan \left (2 e^{\frac {b x}{3}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\sqrt {3} e^{\frac {b x}{3}}+e^{\frac {2 b x}{3}}+1\right )\right )\right )\right )\right )+\frac {e^{\frac {5 b x}{3}} \left (1-e^{2 b x}\right )^2}{6 \left (e^{2 b x}+1\right )^2}\right )}{b}\) |
Input:
Int[E^((5*(a + b*x))/3)*Tanh[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*(ArcTan[E^((b*x)/3)]/3 + (-ArcTan[Sqrt[3] - 2*E^((b*x)/3)] + (Sqrt[3]*Log[1 - Sqrt[3]*E^((b*x)/3) + E^((2*b*x)/3)])/2)/6 + (ArcTan[Sq rt[3] + 2*E^((b*x)/3)] - (Sqrt[3]*Log[1 + Sqrt[3]*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[(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[(-(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[(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 - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/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] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[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 higher order function than in optimal. Order 9 vs. order 3.
Time = 0.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(\frac {3 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{5 b}+\frac {\left (11 \,{\mathrm e}^{2 b x +2 d}+5\right ) {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{3 \left (1+{\mathrm e}^{2 b x +2 d}\right )^{2} b}+\frac {43 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-i\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{18 b}-\frac {43 i \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+i\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}}{18 b}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (104976 b^{4} \textit {\_Z}^{4}-599076 b^{2} \textit {\_Z}^{2}+3418801\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}-\frac {5832 b^{3} \textit {\_R}^{3}}{79507}+\frac {18 b \textit {\_R}}{43}\right )\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}\) | \(166\) |
Input:
int(exp(5/3*b*x+5/3*a)*tanh(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
3/5*exp(5/3*b*x+5/3*a)/b+1/3/(1+exp(2*b*x+2*d))^2/b*(11*exp(2*b*x+2*d)+5)* exp(5/3*b*x+5/3*a)+43/18*I/b*ln(exp(1/3*b*x+1/3*d)-I)*exp(5/3*a-5/3*d)-43/ 18*I/b*ln(exp(1/3*b*x+1/3*d)+I)*exp(5/3*a-5/3*d)+sum(_R*ln(exp(1/3*b*x+1/3 *d)-5832/79507*b^3*_R^3+18/43*b*_R),_R=RootOf(104976*_Z^4*b^4-599076*_Z^2* b^2+3418801))*exp(5/3*a-5/3*d)
Leaf count of result is larger than twice the leaf count of optimal. 6325 vs. \(2 (195) = 390\).
Time = 0.16 (sec) , antiderivative size = 6325, normalized size of antiderivative = 23.34 \[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)*tanh(b*x+d)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=e^{\frac {5 a}{3}} \int e^{\frac {5 b x}{3}} \tanh ^{3}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(5/3*b*x+5/3*a)*tanh(b*x+d)**3,x)
Output:
exp(5*a/3)*Integral(exp(5*b*x/3)*tanh(b*x + d)**3, x)
Time = 0.14 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.72 \[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=\frac {43 \, {\left (\sqrt {3} \log \left (\sqrt {3} e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right ) - \sqrt {3} \log \left (-\sqrt {3} e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (-\frac {2}{3} \, b x - \frac {2}{3} \, d\right )} + 1\right ) + 2 \, \arctan \left (\sqrt {3} + 2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )}\right ) + 2 \, \arctan \left (-\sqrt {3} + 2 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )}\right ) + 4 \, \arctan \left (e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )}\right )\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{36 \, b} + \frac {{\left (73 \, e^{\left (-2 \, b x - 2 \, d\right )} + 34 \, e^{\left (-4 \, b x - 4 \, d\right )} + 9\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{15 \, b {\left (e^{\left (-\frac {5}{3} \, b x - \frac {5}{3} \, d\right )} + 2 \, e^{\left (-\frac {11}{3} \, b x - \frac {11}{3} \, d\right )} + e^{\left (-\frac {17}{3} \, b x - \frac {17}{3} \, d\right )}\right )}} \] Input:
integrate(exp(5/3*b*x+5/3*a)*tanh(b*x+d)^3,x, algorithm="maxima")
Output:
43/36*(sqrt(3)*log(sqrt(3)*e^(-1/3*b*x - 1/3*d) + e^(-2/3*b*x - 2/3*d) + 1 ) - sqrt(3)*log(-sqrt(3)*e^(-1/3*b*x - 1/3*d) + e^(-2/3*b*x - 2/3*d) + 1) + 2*arctan(sqrt(3) + 2*e^(-1/3*b*x - 1/3*d)) + 2*arctan(-sqrt(3) + 2*e^(-1 /3*b*x - 1/3*d)) + 4*arctan(e^(-1/3*b*x - 1/3*d)))*e^(5/3*a - 5/3*d)/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 = 0.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.80 \[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=\frac {215 \, \sqrt {3} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (\sqrt {3} e^{\left (\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right ) - 215 \, \sqrt {3} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} \log \left (-\sqrt {3} e^{\left (\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} + e^{\left (\frac {2}{3} \, b x\right )} + e^{\left (-\frac {2}{3} \, d\right )}\right ) - 430 \, \arctan \left ({\left (\sqrt {3} e^{\left (-\frac {1}{3} \, d\right )} + 2 \, e^{\left (\frac {1}{3} \, b x\right )}\right )} e^{\left (\frac {1}{3} \, d\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} - 430 \, \arctan \left (-{\left (\sqrt {3} e^{\left (-\frac {1}{3} \, d\right )} - 2 \, e^{\left (\frac {1}{3} \, b x\right )}\right )} e^{\left (\frac {1}{3} \, d\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} - 860 \, \arctan \left (e^{\left (\frac {1}{3} \, b x + \frac {1}{3} \, d\right )}\right ) e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )} + \frac {60 \, {\left (11 \, e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 2 \, d\right )} + 5 \, e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, d\right )} + 1\right )}^{2}} + 108 \, e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a\right )}}{180 \, b} \] Input:
integrate(exp(5/3*b*x+5/3*a)*tanh(b*x+d)^3,x, algorithm="giac")
Output:
1/180*(215*sqrt(3)*e^(5/3*a - 5/3*d)*log(sqrt(3)*e^(1/3*b*x - 1/3*d) + e^( 2/3*b*x) + e^(-2/3*d)) - 215*sqrt(3)*e^(5/3*a - 5/3*d)*log(-sqrt(3)*e^(1/3 *b*x - 1/3*d) + e^(2/3*b*x) + e^(-2/3*d)) - 430*arctan((sqrt(3)*e^(-1/3*d) + 2*e^(1/3*b*x))*e^(1/3*d))*e^(5/3*a - 5/3*d) - 430*arctan(-(sqrt(3)*e^(- 1/3*d) - 2*e^(1/3*b*x))*e^(1/3*d))*e^(5/3*a - 5/3*d) - 860*arctan(e^(1/3*b *x + 1/3*d))*e^(5/3*a - 5/3*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 = 5.65 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.86 \[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx =\text {Too large to display} \] Input:
int(exp((5*a)/3 + (5*b*x)/3)*tanh(d + b*x)^3,x)
Output:
(3*exp((5*a)/3 + (5*b*x)/3))/(5*b) - (43*(-exp(10*a - 10*d))^(1/6)*log((18 49*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)*(-exp(10*a)*exp(-10*d))^(1/6))/81))/(18*b) + (43*(-exp( 10*a - 10*d))^(1/6)*log((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)*(-exp(10*a)*exp(-10*d))^(1/6 ))/81))/(18*b) + (11*exp((5*a)/3 + (5*b*x)/3))/(3*b*(exp(2*d + 2*b*x) + 1) ) - (2*exp((5*a)/3 + (5*b*x)/3))/(b*(2*exp(2*d + 2*b*x) + exp(4*d + 4*b*x) + 1)) - (43*log((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((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)*e xp(-10*d))^(1/6))/81)*(-exp(10*a - 10*d))^(1/6)*((3^(1/2)*1i)/2 - 1/2))/(1 8*b) - (43*log((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)*ex p(-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((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)
\[ \int e^{\frac {5}{3} (a+b x)} \tanh ^3(d+b x) \, dx=\int e^{\frac {5 b x}{3}+\frac {5 a}{3}} \tanh \left (b x +d \right )^{3}d x \] Input:
int(exp(5/3*b*x+5/3*a)*tanh(b*x+d)^3,x)
Output:
int(e**((5*a + 5*b*x)/3)*tanh(b*x + d)**3,x)