Integrand size = 39, antiderivative size = 40 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\right ) \] Output:
2/3*arctanh(1/4*(2-5*exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2))
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=-\frac {4}{3} \text {arctanh}\left (1-\frac {1}{2} e^{3 x/4}+\frac {1}{2} \sqrt {-2+e^{3 x/4}+e^{3 x/2}}\right ) \] Input:
Integrate[E^((3*x)/4)/((-2 + E^((3*x)/4))*Sqrt[-2 + E^((3*x)/4) + E^((3*x) /2)]),x]
Output:
(-4*ArcTanh[1 - E^((3*x)/4)/2 + Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)]/2])/3
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2720, 25, 1154, 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.
\(\displaystyle \int \frac {e^{3 x/4}}{\left (e^{3 x/4}-2\right ) \sqrt {e^{3 x/4}+e^{3 x/2}-2}} \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {4}{3} \int -\frac {1}{\left (2-e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}de^{3 x/4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {4}{3} \int \frac {1}{\left (2-e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}}de^{3 x/4}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {8}{3} \int \frac {1}{16-e^{3 x/2}}d\frac {2-5 e^{3 x/4}}{\sqrt {-2+e^{3 x/4}+e^{3 x/2}}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{3} \text {arctanh}\left (\frac {2-5 e^{3 x/4}}{4 \sqrt {e^{3 x/4}+e^{3 x/2}-2}}\right )\) |
Input:
Int[E^((3*x)/4)/((-2 + E^((3*x)/4))*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)]), x]
Output:
(2*ArcTanh[(2 - 5*E^((3*x)/4))/(4*Sqrt[-2 + E^((3*x)/4) + E^((3*x)/2)])])/ 3
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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], 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]]]
\[\int \frac {{\mathrm e}^{\frac {3 x}{4}}}{\left (-2+{\mathrm e}^{\frac {3 x}{4}}\right ) \sqrt {-2+{\mathrm e}^{\frac {3 x}{4}}+{\mathrm e}^{\frac {3 x}{2}}}}d x\]
Input:
int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)
Output:
int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=-\frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )} + 4\right ) + \frac {2}{3} \, \log \left (\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} - e^{\left (\frac {3}{4} \, x\right )}\right ) \] Input:
integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, a lgorithm="fricas")
Output:
-2/3*log(sqrt(e^(3/2*x) + e^(3/4*x) - 2) - e^(3/4*x) + 4) + 2/3*log(sqrt(e ^(3/2*x) + e^(3/4*x) - 2) - e^(3/4*x))
\[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int \frac {e^{\frac {3 x}{4}}}{\left (e^{\frac {3 x}{4}} - 2\right ) \sqrt {e^{\frac {3 x}{4}} + e^{\frac {3 x}{2}} - 2}}\, dx \] Input:
integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))**(1/2),x)
Output:
Integral(exp(3*x/4)/((exp(3*x/4) - 2)*sqrt(exp(3*x/4) + exp(3*x/2) - 2)), x)
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=-\frac {2}{3} \, \log \left (\frac {4 \, \sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2}}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + \frac {8}{{\left | e^{\left (\frac {3}{4} \, x\right )} - 2 \right |}} + 5\right ) \] Input:
integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, a lgorithm="maxima")
Output:
-2/3*log(4*sqrt(e^(3/2*x) + e^(3/4*x) - 2)/abs(e^(3/4*x) - 2) + 8/abs(e^(3 /4*x) - 2) + 5)
\[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int { \frac {e^{\left (\frac {3}{4} \, x\right )}}{\sqrt {e^{\left (\frac {3}{2} \, x\right )} + e^{\left (\frac {3}{4} \, x\right )} - 2} {\left (e^{\left (\frac {3}{4} \, x\right )} - 2\right )}} \,d x } \] Input:
integrate(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x, a lgorithm="giac")
Output:
integrate(e^(3/4*x)/(sqrt(e^(3/2*x) + e^(3/4*x) - 2)*(e^(3/4*x) - 2)), x)
Timed out. \[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int \frac {{\mathrm {e}}^{\frac {3\,x}{4}}}{\left ({\mathrm {e}}^{\frac {3\,x}{4}}-2\right )\,\sqrt {{\mathrm {e}}^{\frac {3\,x}{2}}+{\mathrm {e}}^{\frac {3\,x}{4}}-2}} \,d x \] Input:
int(exp((3*x)/4)/((exp((3*x)/4) - 2)*(exp((3*x)/2) + exp((3*x)/4) - 2)^(1/ 2)),x)
Output:
int(exp((3*x)/4)/((exp((3*x)/4) - 2)*(exp((3*x)/2) + exp((3*x)/4) - 2)^(1/ 2)), x)
\[ \int \frac {e^{3 x/4}}{\left (-2+e^{3 x/4}\right ) \sqrt {-2+e^{3 x/4}+e^{3 x/2}}} \, dx=\int \frac {e^{\frac {3 x}{4}}}{e^{\frac {3 x}{4}} \sqrt {e^{\frac {3 x}{4}}+e^{\frac {3 x}{2}}-2}-2 \sqrt {e^{\frac {3 x}{4}}+e^{\frac {3 x}{2}}-2}}d x \] Input:
int(exp(3/4*x)/(-2+exp(3/4*x))/(-2+exp(3/4*x)+exp(3/2*x))^(1/2),x)
Output:
int(e**((3*x)/4)/(e**((3*x)/4)*sqrt(e**((3*x)/4) + e**((3*x)/2) - 2) - 2*s qrt(e**((3*x)/4) + e**((3*x)/2) - 2)),x)