Integrand size = 8, antiderivative size = 366 \[ \int e^x \tanh (4 x) \, dx=e^x+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {1}{8} \sqrt {2-\sqrt {2}} \log \left (1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )-\frac {1}{8} \sqrt {2-\sqrt {2}} \log \left (1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}\right )+\frac {1}{8} \sqrt {2+\sqrt {2}} \log \left (1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right )-\frac {1}{8} \sqrt {2+\sqrt {2}} \log \left (1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}\right ) \]
exp(x)+1/8*ln(1+exp(2*x)-exp(x)*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)-1/8*l n(1+exp(2*x)+exp(x)*(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/2*arctan((-2*ex p(x)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/2*arctan( (2*exp(x)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)+1/8*ln (1+exp(2*x)-exp(x)*(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)-1/8*ln(1+exp(2*x)+ exp(x)*(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+1/2*arctan((-2*exp(x)+(2+2^(1/ 2))^(1/2))/(2-2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)-1/2*arctan((2*exp(x)+(2+ 2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.07 \[ \int e^x \tanh (4 x) \, dx=e^x-2 e^x \operatorname {Hypergeometric2F1}\left (\frac {1}{8},1,\frac {9}{8},-e^{8 x}\right ) \]
Time = 0.66 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {2720, 25, 913, 757, 1483, 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.
| \(\displaystyle \int e^x \tanh (4 x) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int -\frac {1-e^{8 x}}{e^{8 x}+1}de^x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1-e^{8 x}}{1+e^{8 x}}de^x\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle e^x-2 \int \frac {1}{1+e^{8 x}}de^x\) |
| \(\Big \downarrow \) 757 |
| \(\displaystyle e^x-2 \left (\frac {\int \frac {\sqrt {2}-e^{2 x}}{1-\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}+e^{2 x}}{1+\sqrt {2} e^{2 x}+e^{4 x}}de^x}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle e^x-2 \left (\frac {\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}+\left (1-\sqrt {2}\right ) e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )}-\left (1-\sqrt {2}\right ) e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}-\left (1+\sqrt {2}\right ) e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (2+\sqrt {2}\right )}+\left (1+\sqrt {2}\right ) e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle e^x-2 \left (\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1-\sqrt {2}\right ) \int -\frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1+\sqrt {2}\right ) \int -\frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e^x-2 \left (\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {2}} \int \frac {1}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {2}} \int \frac {1}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle e^x-2 \left (\frac {\frac {-\sqrt {2+\sqrt {2}} \int \frac {1}{-2-\sqrt {2}-e^{2 x}}d\left (-\sqrt {2-\sqrt {2}}+2 e^x\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {-\sqrt {2+\sqrt {2}} \int \frac {1}{-2-\sqrt {2}-e^{2 x}}d\left (\sqrt {2-\sqrt {2}}+2 e^x\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x-\sqrt {2-\sqrt {2}} \int \frac {1}{-2+\sqrt {2}-e^{2 x}}d\left (-\sqrt {2+\sqrt {2}}+2 e^x\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x-\sqrt {2-\sqrt {2}} \int \frac {1}{-2+\sqrt {2}-e^{2 x}}d\left (\sqrt {2+\sqrt {2}}+2 e^x\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle e^x-2 \left (\frac {\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}-2 e^x}{1-\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {\sqrt {2-\sqrt {2}}+2 e^x}{1+\sqrt {2-\sqrt {2}} e^x+e^{2 x}}de^x}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}-2 e^x}{1-\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {\sqrt {2+\sqrt {2}}+2 e^x}{1+\sqrt {2+\sqrt {2}} e^x+e^{2 x}}de^x+\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle e^x-2 \left (\frac {\frac {\arctan \left (\frac {2 e^x-\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (-\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2-\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{2} \left (1-\sqrt {2}\right ) \log \left (\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2-\sqrt {2}}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\frac {2 e^x-\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (-\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2+\sqrt {2}}}+\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{2} \left (1+\sqrt {2}\right ) \log \left (\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )}{2 \sqrt {2+\sqrt {2}}}}{2 \sqrt {2}}\right )\) |
E^x - 2*(((ArcTan[(-Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]] + ((1 - Sqrt[2])*Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)])/2)/(2*Sqrt[2 - Sqrt[2]] ) + (ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]] - ((1 - Sqrt[2] )*Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)])/2)/(2*Sqrt[2 - Sqrt[2]]))/(2*S qrt[2]) + ((ArcTan[(-Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]] - ((1 + Sqrt[2])*Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)])/2)/(2*Sqrt[2 + Sqrt[2] ]) + (ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]] + ((1 + Sqrt[2 ])*Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)])/2)/(2*Sqrt[2 + Sqrt[2]]))/(2* Sqrt[2]))
3.3.23.3.1 Defintions of rubi rules used
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_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b, 4]]}, Simp[r/(2*Sqrt[2]*a) Int[(Sqrt[2]*r - s*x^(n/4))/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] + Simp[r/(2*S qrt[2]*a) Int[(Sqrt[2]*r + s*x^(n/4))/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^ (n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && GtQ[a/b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
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[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
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 = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.07
| method | result | size |
| risch | \({\mathrm e}^{x}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (65536 \textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-4 \textit {\_R} \right )\right )\) | \(24\) |
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.47 \[ \int e^x \tanh (4 x) \, dx=-\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} \log \left (-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) - \frac {1}{4} \, \left (-1\right )^{\frac {1}{8}} \log \left (\left (-1\right )^{\frac {1}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{4} i \, \left (-1\right )^{\frac {1}{8}} \log \left (i \, \left (-1\right )^{\frac {1}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \frac {1}{4} i \, \left (-1\right )^{\frac {1}{8}} \log \left (-i \, \left (-1\right )^{\frac {1}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \frac {1}{4} \, \left (-1\right )^{\frac {1}{8}} \log \left (-\left (-1\right )^{\frac {1}{8}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \cosh \left (x\right ) + \sinh \left (x\right ) \]
-(1/8*I + 1/8)*sqrt(2)*(-1)^(1/8)*log((I + 1)*sqrt(2)*(-1)^(1/8) + 2*cosh( x) + 2*sinh(x)) + (1/8*I - 1/8)*sqrt(2)*(-1)^(1/8)*log(-(I - 1)*sqrt(2)*(- 1)^(1/8) + 2*cosh(x) + 2*sinh(x)) - (1/8*I - 1/8)*sqrt(2)*(-1)^(1/8)*log(( I - 1)*sqrt(2)*(-1)^(1/8) + 2*cosh(x) + 2*sinh(x)) + (1/8*I + 1/8)*sqrt(2) *(-1)^(1/8)*log(-(I + 1)*sqrt(2)*(-1)^(1/8) + 2*cosh(x) + 2*sinh(x)) - 1/4 *(-1)^(1/8)*log((-1)^(1/8) + cosh(x) + sinh(x)) - 1/4*I*(-1)^(1/8)*log(I*( -1)^(1/8) + cosh(x) + sinh(x)) + 1/4*I*(-1)^(1/8)*log(-I*(-1)^(1/8) + cosh (x) + sinh(x)) + 1/4*(-1)^(1/8)*log(-(-1)^(1/8) + cosh(x) + sinh(x)) + cos h(x) + sinh(x)
\[ \int e^x \tanh (4 x) \, dx=\int e^{x} \tanh {\left (4 x \right )}\, dx \]
\[ \int e^x \tanh (4 x) \, dx=\int { e^{x} \tanh \left (4 \, x\right ) \,d x } \]
Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.69 \[ \int e^x \tanh (4 x) \, dx=-\frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + e^{x} \]
-1/4*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*e^x)/sqrt(-sqrt(2) + 2)) - 1/4*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(sqrt(2) + 2) - 2*e^x)/sqrt(-sq rt(2) + 2)) - 1/4*sqrt(sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*e^x)/sq rt(sqrt(2) + 2)) - 1/4*sqrt(sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*e ^x)/sqrt(sqrt(2) + 2)) - 1/8*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/8*sqrt(sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*e^x + e^(2*x) + 1) - 1/8*sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/8*sqrt(-sqrt(2) + 2)*log(-sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) + e^x
Time = 4.22 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.25 \[ \int e^x \tanh (4 x) \, dx={\mathrm {e}}^x-\ln \left (2\,{\mathrm {e}}^x+\sqrt {\sqrt {2}+2}+\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )+\ln \left (2\,{\mathrm {e}}^x+\sqrt {2-\sqrt {2}}-\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )\,\left (-\frac {\sqrt {2-\sqrt {2}}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )+\ln \left (2\,{\mathrm {e}}^x-\sqrt {\sqrt {2}+2}-\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )-\ln \left (2\,{\mathrm {e}}^x-\sqrt {2-\sqrt {2}}+\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}\right )\,\left (-\frac {\sqrt {2-\sqrt {2}}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )+\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^x+\sqrt {2}\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (-4-4{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^x+\sqrt {2}\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (-4+4{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^x+\sqrt {2}\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (4-4{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^x+\sqrt {2}\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (4+4{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {\sqrt {2}+2}}{8}+\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \]
exp(x) - log(2*exp(x) + (2^(1/2) + 2)^(1/2) + (2 - 2^(1/2))^(1/2)*1i)*((2^ (1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8) + log(2*exp(x) - (2^(1/2) + 2)^(1/2)*1i + (2 - 2^(1/2))^(1/2))*(((2^(1/2) + 2)^(1/2)*1i)/8 - (2 - 2 ^(1/2))^(1/2)/8) + log(2*exp(x) - (2^(1/2) + 2)^(1/2) - (2 - 2^(1/2))^(1/2 )*1i)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8) - log(2*exp(x) + (2^(1/2) + 2)^(1/2)*1i - (2 - 2^(1/2))^(1/2))*(((2^(1/2) + 2)^(1/2)*1i)/ 8 - (2 - 2^(1/2))^(1/2)/8) + 2^(1/2)*log(2*exp(x) - 2^(1/2)*((2^(1/2) + 2) ^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(4 + 4i))*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 + 1i/2) + 2^(1/2)*log(2*exp(x) - 2^(1/2)* ((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(4 - 4i))*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 - 1i/2) - 2^(1/2)*log(2*exp (x) + 2^(1/2)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(4 - 4i ))*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 - 1i/2) - 2^( 1/2)*log(2*exp(x) + 2^(1/2)*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)* 1i)/8)*(4 + 4i))*((2^(1/2) + 2)^(1/2)/8 + ((2 - 2^(1/2))^(1/2)*1i)/8)*(1/2 + 1i/2)