Integrand size = 10, antiderivative size = 239 \[ \int \frac {1}{a-b x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}} \]
1/4*arctan(b^(1/8)*x/a^(1/8))/a^(7/8)/b^(1/8)+1/4*arctanh(b^(1/8)*x/a^(1/8 ))/a^(7/8)/b^(1/8)+1/8*arctan(-1+b^(1/8)*x*2^(1/2)/a^(1/8))/a^(7/8)/b^(1/8 )*2^(1/2)+1/8*arctan(1+b^(1/8)*x*2^(1/2)/a^(1/8))/a^(7/8)/b^(1/8)*2^(1/2)- 1/16*ln(a^(1/4)+b^(1/4)*x^2-a^(1/8)*b^(1/8)*x*2^(1/2))/a^(7/8)/b^(1/8)*2^( 1/2)+1/16*ln(a^(1/4)+b^(1/4)*x^2+a^(1/8)*b^(1/8)*x*2^(1/2))/a^(7/8)/b^(1/8 )*2^(1/2)
Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.83 \[ \int \frac {1}{a-b x^8} \, dx=\frac {4 \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )-2 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )+2 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )-2 \log \left (\sqrt [8]{a}-\sqrt [8]{b} x\right )+2 \log \left (\sqrt [8]{a}+\sqrt [8]{b} x\right )-\sqrt {2} \log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )+\sqrt {2} \log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{16 a^{7/8} \sqrt [8]{b}} \]
(4*ArcTan[(b^(1/8)*x)/a^(1/8)] - 2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/ a^(1/8)] + 2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/a^(1/8)] - 2*Log[a^(1/ 8) - b^(1/8)*x] + 2*Log[a^(1/8) + b^(1/8)*x] - Sqrt[2]*Log[a^(1/4) - Sqrt[ 2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^2] + Sqrt[2]*Log[a^(1/4) + Sqrt[2]*a^(1/8 )*b^(1/8)*x + b^(1/4)*x^2])/(16*a^(7/8)*b^(1/8))
Time = 0.47 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {758, 755, 756, 218, 221, 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.
| \(\displaystyle \int \frac {1}{a-b x^8} \, dx\) |
| \(\Big \downarrow \) 758 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {a}-\sqrt {b} x^4}dx}{2 \sqrt {a}}+\frac {\int \frac {1}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {a}-\sqrt {b} x^4}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{a}}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\sqrt [4]{a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{a}}+\frac {\int \frac {1}{\sqrt [4]{b} x^2+\sqrt [4]{a}}dx}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{a}}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\sqrt [4]{a}-\sqrt [4]{b} x^2}dx}{2 \sqrt [4]{a}}+\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{a}}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{b} x^2+\sqrt [4]{a}}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}}dx}{2 \sqrt [4]{b}}}{2 \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {b} x^4+\sqrt {a}}dx}{2 \sqrt [4]{a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{a}-2 \sqrt [8]{b} x}{\sqrt [8]{b} \left (x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{a}\right )}{\sqrt [8]{b} \left (x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{a}-2 \sqrt [8]{b} x}{x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{b} x+\sqrt [8]{a}}{x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+\frac {\sqrt [4]{a}}{\sqrt [4]{b}}}dx}{2 \sqrt [8]{a} \sqrt [4]{b}}}{2 \sqrt [4]{a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{2 a^{3/8} \sqrt [8]{b}}}{2 \sqrt {a}}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{b}}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}\) |
(ArcTan[(b^(1/8)*x)/a^(1/8)]/(2*a^(3/8)*b^(1/8)) + ArcTanh[(b^(1/8)*x)/a^( 1/8)]/(2*a^(3/8)*b^(1/8)))/(2*Sqrt[a]) + ((-(ArcTan[1 - (Sqrt[2]*b^(1/8)*x )/a^(1/8)]/(Sqrt[2]*a^(1/8)*b^(1/8))) + ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/a^( 1/8)]/(Sqrt[2]*a^(1/8)*b^(1/8)))/(2*a^(1/4)) + (-1/2*Log[a^(1/4) - Sqrt[2] *a^(1/8)*b^(1/8)*x + b^(1/4)*x^2]/(Sqrt[2]*a^(1/8)*b^(1/8)) + Log[a^(1/4) + Sqrt[2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^2]/(2*Sqrt[2]*a^(1/8)*b^(1/8)))/(2 *a^(1/4)))/(2*Sqrt[a])
3.15.70.3.1 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_)^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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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_)^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[((a_) + (b_.)*(x_)^(n_))^(-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^(n/2)), x], x] + Simp[r/(2*a) Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 1] && !GtQ[a/b, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12
| method | result | size |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\) | \(29\) |
| risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\) | \(29\) |
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97 \[ \int \frac {1}{a-b x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} i \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (i \, a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} i \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-i \, a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) \]
(1/16*I + 1/16)*sqrt(2)*(1/(a^7*b))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*(1/( a^7*b))^(1/8) + x) - (1/16*I - 1/16)*sqrt(2)*(1/(a^7*b))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a*(1/(a^7*b))^(1/8) + x) + (1/16*I - 1/16)*sqrt(2)*(1/(a^7 *b))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a*(1/(a^7*b))^(1/8) + x) - (1/16*I + 1/16)*sqrt(2)*(1/(a^7*b))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a*(1/(a^7*b))^( 1/8) + x) + 1/8*(1/(a^7*b))^(1/8)*log(a*(1/(a^7*b))^(1/8) + x) + 1/8*I*(1/ (a^7*b))^(1/8)*log(I*a*(1/(a^7*b))^(1/8) + x) - 1/8*I*(1/(a^7*b))^(1/8)*lo g(-I*a*(1/(a^7*b))^(1/8) + x) - 1/8*(1/(a^7*b))^(1/8)*log(-a*(1/(a^7*b))^( 1/8) + x)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.09 \[ \int \frac {1}{a-b x^8} \, dx=- \operatorname {RootSum} {\left (16777216 t^{8} a^{7} b - 1, \left ( t \mapsto t \log {\left (- 8 t a + x \right )} \right )\right )} \]
\[ \int \frac {1}{a-b x^8} \, dx=\int { -\frac {1}{b x^{8} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (154) = 308\).
Time = 0.29 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.93 \[ \int \frac {1}{a-b x^8} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} \]
1/4*(-a/b)^(1/8)*arctan((2*x + sqrt(-sqrt(2) + 2)*(-a/b)^(1/8))/(sqrt(sqrt (2) + 2)*(-a/b)^(1/8)))/(a*sqrt(-2*sqrt(2) + 4)) + 1/4*(-a/b)^(1/8)*arctan ((2*x - sqrt(-sqrt(2) + 2)*(-a/b)^(1/8))/(sqrt(sqrt(2) + 2)*(-a/b)^(1/8))) /(a*sqrt(-2*sqrt(2) + 4)) + 1/4*(-a/b)^(1/8)*arctan((2*x + sqrt(sqrt(2) + 2)*(-a/b)^(1/8))/(sqrt(-sqrt(2) + 2)*(-a/b)^(1/8)))/(a*sqrt(2*sqrt(2) + 4) ) + 1/4*(-a/b)^(1/8)*arctan((2*x - sqrt(sqrt(2) + 2)*(-a/b)^(1/8))/(sqrt(- sqrt(2) + 2)*(-a/b)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) + 1/8*(-a/b)^(1/8)*log (x^2 + x*sqrt(sqrt(2) + 2)*(-a/b)^(1/8) + (-a/b)^(1/4))/(a*sqrt(-2*sqrt(2) + 4)) - 1/8*(-a/b)^(1/8)*log(x^2 - x*sqrt(sqrt(2) + 2)*(-a/b)^(1/8) + (-a /b)^(1/4))/(a*sqrt(-2*sqrt(2) + 4)) + 1/8*(-a/b)^(1/8)*log(x^2 + x*sqrt(-s qrt(2) + 2)*(-a/b)^(1/8) + (-a/b)^(1/4))/(a*sqrt(2*sqrt(2) + 4)) - 1/8*(-a /b)^(1/8)*log(x^2 - x*sqrt(-sqrt(2) + 2)*(-a/b)^(1/8) + (-a/b)^(1/4))/(a*s qrt(2*sqrt(2) + 4))
Time = 6.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.39 \[ \int \frac {1}{a-b x^8} \, dx=\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{7/8}\,b^{1/8}}-\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{7/8}\,b^{1/8}} \]