Integrand size = 17, antiderivative size = 128 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}{-\sqrt {b}+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{b}} \]
-1/4*arctan(2^(1/2)*b^(1/4)*(a*x^4-b)^(1/4)/(-b^(1/2)+(a*x^4-b)^(1/2)))*2^ (1/2)/b^(1/4)-1/4*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*(a*x^4-b)^(1/2)*2^(1/2) /b^(1/4))/(a*x^4-b)^(1/4))*2^(1/2)/b^(1/4)
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\arctan \left (\frac {-\sqrt {b}+\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^4}}{\sqrt {b}+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{b}} \]
(ArcTan[(-Sqrt[b] + Sqrt[-b + a*x^4])/(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4)) ] - ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^ 4])])/(2*Sqrt[2]*b^(1/4))
Time = 0.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.45, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {798, 73, 27, 826, 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.
| \(\displaystyle \int \frac {1}{x \sqrt [4]{a x^4-b}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{x^4 \sqrt [4]{a x^4-b}}dx^4\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\int \frac {a x^8}{x^{16}+b}d\sqrt [4]{a x^4-b}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x^8}{b+x^{16}}d\sqrt [4]{a x^4-b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{2} \int \frac {x^8+\sqrt {b}}{x^{16}+b}d\sqrt [4]{a x^4-b}-\frac {1}{2} \int \frac {\sqrt {b}-x^8}{x^{16}+b}d\sqrt [4]{a x^4-b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^8+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}+\frac {1}{2} \int \frac {1}{x^8+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^8}{x^{16}+b}d\sqrt [4]{a x^4-b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{-x^8-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^8-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^8}{x^{16}+b}d\sqrt [4]{a x^4-b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^8}{x^{16}+b}d\sqrt [4]{a x^4-b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^4-b}}{x^8+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^4-b}\right )}{x^8+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^4-b}}{x^8+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^4-b}\right )}{x^8+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a x^4-b}}{x^8+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {\sqrt [4]{b}+\sqrt {2} \sqrt [4]{a x^4-b}}{x^8+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}}d\sqrt [4]{a x^4-b}}{2 \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^4-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}+\sqrt {b}+x^8\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^4-b}+\sqrt {b}+x^8\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\) |
(-(ArcTan[1 - (Sqrt[2]*(-b + a*x^4)^(1/4))/b^(1/4)]/(Sqrt[2]*b^(1/4))) + A rcTan[1 + (Sqrt[2]*(-b + a*x^4)^(1/4))/b^(1/4)]/(Sqrt[2]*b^(1/4)))/2 + (Lo g[Sqrt[b] + x^8 - Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[Sqrt[b] + x^8 + Sqrt[2]*b^(1/4)*(-b + a*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/ 4)))/2
3.19.61.3.1 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[((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[(-(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_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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_) + (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]
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.05
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {a \,x^{4}-b}-b^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{4}-b}+b^{\frac {1}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{4}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \left (a \,x^{4}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )\right )}{8 b^{\frac {1}{4}}}\) | \(134\) |
1/8/b^(1/4)*2^(1/2)*(ln(((a*x^4-b)^(1/2)-b^(1/4)*(a*x^4-b)^(1/4)*2^(1/2)+b ^(1/2))/((a*x^4-b)^(1/2)+b^(1/4)*(a*x^4-b)^(1/4)*2^(1/2)+b^(1/2)))+2*arcta n((2^(1/2)*(a*x^4-b)^(1/4)+b^(1/4))/b^(1/4))-2*arctan((-2^(1/2)*(a*x^4-b)^ (1/4)+b^(1/4))/b^(1/4)))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {1}{4} \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} i \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (i \, b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} i \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (-i \, b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \left (-\frac {1}{b}\right )^{\frac {1}{4}} \log \left (-b \left (-\frac {1}{b}\right )^{\frac {3}{4}} + {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right ) \]
1/4*(-1/b)^(1/4)*log(b*(-1/b)^(3/4) + (a*x^4 - b)^(1/4)) - 1/4*I*(-1/b)^(1 /4)*log(I*b*(-1/b)^(3/4) + (a*x^4 - b)^(1/4)) + 1/4*I*(-1/b)^(1/4)*log(-I* b*(-1/b)^(3/4) + (a*x^4 - b)^(1/4)) - 1/4*(-1/b)^(1/4)*log(-b*(-1/b)^(3/4) + (a*x^4 - b)^(1/4))
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{4}}} \right )}}{4 \sqrt [4]{a} x \Gamma \left (\frac {5}{4}\right )} \]
-gamma(1/4)*hyper((1/4, 1/4), (5/4,), b*exp_polar(2*I*pi)/(a*x**4))/(4*a** (1/4)*x*gamma(5/4))
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} \]
1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^4 - b)^(1/4))/b^( 1/4))/b^(1/4) + 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^ 4 - b)^(1/4))/b^(1/4))/b^(1/4) - 1/8*sqrt(2)*log(sqrt(2)*(a*x^4 - b)^(1/4) *b^(1/4) + sqrt(a*x^4 - b) + sqrt(b))/b^(1/4) + 1/8*sqrt(2)*log(-sqrt(2)*( a*x^4 - b)^(1/4)*b^(1/4) + sqrt(a*x^4 - b) + sqrt(b))/b^(1/4)
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{4 \, b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{4} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{4} - b} + \sqrt {b}\right )}{8 \, b^{\frac {1}{4}}} \]
1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^4 - b)^(1/4))/b^( 1/4))/b^(1/4) + 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a*x^ 4 - b)^(1/4))/b^(1/4))/b^(1/4) - 1/8*sqrt(2)*log(sqrt(2)*(a*x^4 - b)^(1/4) *b^(1/4) + sqrt(a*x^4 - b) + sqrt(b))/b^(1/4) + 1/8*sqrt(2)*log(-sqrt(2)*( a*x^4 - b)^(1/4)*b^(1/4) + sqrt(a*x^4 - b) + sqrt(b))/b^(1/4)
Time = 5.81 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.36 \[ \int \frac {1}{x \sqrt [4]{-b+a x^4}} \, dx=\frac {\mathrm {atan}\left (\frac {{\left (a\,x^4-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )-\mathrm {atanh}\left (\frac {{\left (a\,x^4-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{2\,{\left (-b\right )}^{1/4}} \]