Integrand size = 15, antiderivative size = 54 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \sqrt [4]{1-2 e^{x/3}}-6 \arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \] Output:
12*(1-2*exp(1/3*x))^(1/4)-6*arctan((1-2*exp(1/3*x))^(1/4))-6*arctanh((1-2* exp(1/3*x))^(1/4))
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \sqrt [4]{1-2 e^{x/3}}-6 \arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )-6 \text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right ) \] Input:
Integrate[(1 - 2*E^(x/3))^(1/4),x]
Output:
12*(1 - 2*E^(x/3))^(1/4) - 6*ArcTan[(1 - 2*E^(x/3))^(1/4)] - 6*ArcTanh[(1 - 2*E^(x/3))^(1/4)]
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2720, 60, 73, 756, 216, 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.
\(\displaystyle \int \sqrt [4]{1-2 e^{x/3}} \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle 3 \int e^{-x/3} \sqrt [4]{1-2 e^{x/3}}de^{x/3}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 3 \left (\int \frac {e^{-x/3}}{\left (1-2 e^{x/3}\right )^{3/4}}de^{x/3}+4 \sqrt [4]{1-2 e^{x/3}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 3 \left (4 \sqrt [4]{1-2 e^{x/3}}-2 \int \frac {1}{\frac {1}{2}-\frac {1}{2} e^{4 x/3}}d\sqrt [4]{1-2 e^{x/3}}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle 3 \left (4 \sqrt [4]{1-2 e^{x/3}}-2 \left (\int \frac {1}{1-e^{2 x/3}}d\sqrt [4]{1-2 e^{x/3}}+\int \frac {1}{1+e^{2 x/3}}d\sqrt [4]{1-2 e^{x/3}}\right )\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 3 \left (4 \sqrt [4]{1-2 e^{x/3}}-2 \left (\int \frac {1}{1-e^{2 x/3}}d\sqrt [4]{1-2 e^{x/3}}+\arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 3 \left (4 \sqrt [4]{1-2 e^{x/3}}-2 \left (\arctan \left (\sqrt [4]{1-2 e^{x/3}}\right )+\text {arctanh}\left (\sqrt [4]{1-2 e^{x/3}}\right )\right )\right )\) |
Input:
Int[(1 - 2*E^(x/3))^(1/4),x]
Output:
3*(4*(1 - 2*E^(x/3))^(1/4) - 2*(ArcTan[(1 - 2*E^(x/3))^(1/4)] + ArcTanh[(1 - 2*E^(x/3))^(1/4)]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[(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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
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]]]
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(12 \left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+1\right )-6 \arctan \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}\right )\) | \(57\) |
default | \(12 \left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}-1\right )-3 \ln \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}+1\right )-6 \arctan \left (\left (1-2 \,{\mathrm e}^{\frac {x}{3}}\right )^{\frac {1}{4}}\right )\) | \(57\) |
Input:
int((1-2*exp(1/3*x))^(1/4),x,method=_RETURNVERBOSE)
Output:
12*(1-2*exp(1/3*x))^(1/4)+3*ln((1-2*exp(1/3*x))^(1/4)-1)-3*ln((1-2*exp(1/3 *x))^(1/4)+1)-6*arctan((1-2*exp(1/3*x))^(1/4))
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1\right ) \] Input:
integrate((1-2*exp(1/3*x))^(1/4),x, algorithm="fricas")
Output:
12*(-2*e^(1/3*x) + 1)^(1/4) - 6*arctan((-2*e^(1/3*x) + 1)^(1/4)) - 3*log(( -2*e^(1/3*x) + 1)^(1/4) + 1) + 3*log((-2*e^(1/3*x) + 1)^(1/4) - 1)
Time = 0.53 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \sqrt [4]{1 - 2 e^{\frac {x}{3}}} + 3 \log {\left (\sqrt [4]{1 - 2 e^{\frac {x}{3}}} - 1 \right )} - 3 \log {\left (\sqrt [4]{1 - 2 e^{\frac {x}{3}}} + 1 \right )} - 6 \operatorname {atan}{\left (\sqrt [4]{1 - 2 e^{\frac {x}{3}}} \right )} \] Input:
integrate((1-2*exp(1/3*x))**(1/4),x)
Output:
12*(1 - 2*exp(x/3))**(1/4) + 3*log((1 - 2*exp(x/3))**(1/4) - 1) - 3*log((1 - 2*exp(x/3))**(1/4) + 1) - 6*atan((1 - 2*exp(x/3))**(1/4))
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1\right ) \] Input:
integrate((1-2*exp(1/3*x))^(1/4),x, algorithm="maxima")
Output:
12*(-2*e^(1/3*x) + 1)^(1/4) - 6*arctan((-2*e^(1/3*x) + 1)^(1/4)) - 3*log(( -2*e^(1/3*x) + 1)^(1/4) + 1) + 3*log((-2*e^(1/3*x) + 1)^(1/4) - 1)
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=12 \, {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 6 \, \arctan \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}}\right ) - 3 \, \log \left ({\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} + 1\right ) + 3 \, \log \left ({\left | {\left (-2 \, e^{\left (\frac {1}{3} \, x\right )} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:
integrate((1-2*exp(1/3*x))^(1/4),x, algorithm="giac")
Output:
12*(-2*e^(1/3*x) + 1)^(1/4) - 6*arctan((-2*e^(1/3*x) + 1)^(1/4)) - 3*log(( -2*e^(1/3*x) + 1)^(1/4) + 1) + 3*log(abs((-2*e^(1/3*x) + 1)^(1/4) - 1))
Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=\frac {12\,{\left (2-4\,{\mathrm {e}}^{x/3}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},-\frac {1}{4};\ \frac {3}{4};\ \frac {{\mathrm {e}}^{-\frac {x}{3}}}{2}\right )}{{\left (2-{\mathrm {e}}^{-\frac {x}{3}}\right )}^{1/4}} \] Input:
int((1 - 2*exp(x/3))^(1/4),x)
Output:
(12*(2 - 4*exp(x/3))^(1/4)*hypergeom([-1/4, -1/4], 3/4, exp(-x/3)/2))/(2 - exp(-x/3))^(1/4)
\[ \int \sqrt [4]{1-2 e^{x/3}} \, dx=\int \left (-2 e^{\frac {x}{3}}+1\right )^{\frac {1}{4}}d x \] Input:
int((1-2*exp(1/3*x))^(1/4),x)
Output:
int(( - 2*e**(x/3) + 1)**(1/4),x)