Integrand size = 24, antiderivative size = 216 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt {2} \sqrt {e}}+\frac {\log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{2 \sqrt {2} \sqrt {e}} \]
-1/2*arctan(1-2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)/e^(1/2))*2^(1/2)/e^(1/2)+ 1/2*arctan(1+2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)/e^(1/2))*2^(1/2)/e^(1/2)-1 /4*ln(e^(1/2)-2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)+x*e^(1/2)/(-x^2+1)^(1/2)) *2^(1/2)/e^(1/2)+1/4*ln(e^(1/2)+2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)+x*e^(1/ 2)/(-x^2+1)^(1/2))*2^(1/2)/e^(1/2)
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=\frac {\sqrt {x} \left (\arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{-x+\sqrt {1-x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{x+\sqrt {1-x^2}}\right )\right )}{\sqrt {2} \sqrt {e x}} \]
(Sqrt[x]*(ArcTan[(Sqrt[2]*Sqrt[x]*(1 - x^2)^(1/4))/(-x + Sqrt[1 - x^2])] + ArcTanh[(Sqrt[2]*Sqrt[x]*(1 - x^2)^(1/4))/(x + Sqrt[1 - x^2])]))/(Sqrt[2] *Sqrt[e*x])
Time = 0.36 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {135, 266, 770, 755, 27, 1476, 1082, 217, 1479, 25, 27, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x} \sqrt [4]{x+1} \sqrt {e x}} \, dx\) |
\(\Big \downarrow \) 135 |
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x^2} \sqrt {e x}}dx\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 \int \frac {1}{\sqrt [4]{1-x^2}}d\sqrt {e x}}{e}\) |
\(\Big \downarrow \) 770 |
\(\displaystyle \frac {2 \int \frac {1}{x^2+1}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{e}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {e^2 (e-e x)}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 e}+\frac {\int \frac {e^2 (x e+e)}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 e}\right )}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \int \frac {x e+e}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{e}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \left (\frac {1}{2} \int \frac {1}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} \int \frac {1}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )\right )}{e}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{e}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{e}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-\frac {2 \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-\frac {2 \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-\frac {2 \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{e}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {2 \left (\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\log \left (\frac {\sqrt {2} \sqrt {e} \sqrt {e x}}{\sqrt [4]{1-x^2}}+e x+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {e x}}{\sqrt [4]{1-x^2}}+e x+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{e}\) |
(2*((e*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[ 2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/( Sqrt[2]*Sqrt[e])))/2 + (e*(-1/2*Log[e + e*x - (Sqrt[2]*Sqrt[e]*Sqrt[e*x])/ (1 - x^2)^(1/4)]/(Sqrt[2]*Sqrt[e]) + Log[e + e*x + (Sqrt[2]*Sqrt[e]*Sqrt[e *x])/(1 - x^2)^(1/4)]/(2*Sqrt[2]*Sqrt[e])))/2))/e
3.10.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
\[\int \frac {1}{\left (1-x \right )^{\frac {1}{4}} \sqrt {e x}\, \left (1+x \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} \, \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + {\left (e x^{2} - e\right )} \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}}}{x^{2} - 1}\right ) + \frac {1}{2} \, \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - {\left (e x^{2} - e\right )} \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}}}{x^{2} - 1}\right ) + \frac {1}{2} i \, \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - {\left (i \, e x^{2} - i \, e\right )} \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}}}{x^{2} - 1}\right ) - \frac {1}{2} i \, \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - {\left (-i \, e x^{2} + i \, e\right )} \left (-\frac {1}{e^{2}}\right )^{\frac {1}{4}}}{x^{2} - 1}\right ) \]
-1/2*(-1/e^2)^(1/4)*log((sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4) + (e*x^2 - e)*(-1/e^2)^(1/4))/(x^2 - 1)) + 1/2*(-1/e^2)^(1/4)*log((sqrt(e*x)*(x + 1) ^(3/4)*(-x + 1)^(3/4) - (e*x^2 - e)*(-1/e^2)^(1/4))/(x^2 - 1)) + 1/2*I*(-1 /e^2)^(1/4)*log((sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4) - (I*e*x^2 - I*e)* (-1/e^2)^(1/4))/(x^2 - 1)) - 1/2*I*(-1/e^2)^(1/4)*log((sqrt(e*x)*(x + 1)^( 3/4)*(-x + 1)^(3/4) - (-I*e*x^2 + I*e)*(-1/e^2)^(1/4))/(x^2 - 1))
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=- \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {3}{8}, \frac {7}{8} & \frac {1}{2}, \frac {3}{4}, 1, 1 \\0, \frac {3}{8}, \frac {1}{2}, \frac {7}{8}, 1, 0 & \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \sqrt {e} \Gamma \left (\frac {1}{4}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{8}, \frac {1}{4}, \frac {3}{8}, \frac {3}{4}, 1 & \\- \frac {1}{8}, \frac {3}{8} & - \frac {1}{4}, 0, \frac {1}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \sqrt {e} \Gamma \left (\frac {1}{4}\right )} \]
-I*meijerg(((3/8, 7/8), (1/2, 3/4, 1, 1)), ((0, 3/8, 1/2, 7/8, 1, 0), ()), exp_polar(-2*I*pi)/x**2)*exp(I*pi/4)/(4*pi*sqrt(e)*gamma(1/4)) - meijerg( ((-1/4, -1/8, 1/4, 3/8, 3/4, 1), ()), ((-1/8, 3/8), (-1/4, 0, 1/4, 0)), x* *(-2))/(4*pi*sqrt(e)*gamma(1/4))
\[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\sqrt {e x} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\sqrt {e x} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=\int \frac {1}{\sqrt {e\,x}\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]