Integrand size = 30, antiderivative size = 116 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]
Time = 0.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a+a^2-a b-\text {$\#$1}^4-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}-2 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]
-1/4*RootSum[a + a^2 - a*b - #1^4 - 2*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*L og[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*# 1]*#1^4)/(-#1 - 2*a*#1 + 2*#1^5) & ]/b
Leaf count is larger than twice the leaf count of optimal. \(445\) vs. \(2(116)=232\).
Time = 0.58 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1758, 25, 902, 756, 218, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{a x^4-b} \left (a x^8-b-x^4\right )} \, dx\) |
\(\Big \downarrow \) 1758 |
\(\displaystyle \frac {2 a \int -\frac {1}{\left (-2 a x^4+\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {4 a b+1}}-\frac {2 a \int -\frac {1}{\left (-2 a x^4-\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {4 a b+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 a \int \frac {1}{\left (-2 a x^4-\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {4 a b+1}}-\frac {2 a \int \frac {1}{\left (-2 a x^4+\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {4 a b+1}}\) |
\(\Big \downarrow \) 902 |
\(\displaystyle \frac {2 a \int \frac {1}{-\frac {\left (a \left (1-\sqrt {4 a b+1}\right )-2 a b\right ) x^4}{a x^4-b}-\sqrt {4 a b+1}+1}d\frac {x}{\sqrt [4]{a x^4-b}}}{\sqrt {4 a b+1}}-\frac {2 a \int \frac {1}{-\frac {\left (a \left (\sqrt {4 a b+1}+1\right )-2 a b\right ) x^4}{a x^4-b}+\sqrt {4 a b+1}+1}d\frac {x}{\sqrt [4]{a x^4-b}}}{\sqrt {4 a b+1}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {2 a \left (\frac {\int \frac {1}{\sqrt {1-\sqrt {4 a b+1}}-\frac {\sqrt {a} \sqrt {-2 b-\sqrt {4 a b+1}+1} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {1-\sqrt {4 a b+1}}}+\frac {\int \frac {1}{\frac {\sqrt {a} \sqrt {-2 b-\sqrt {4 a b+1}+1} x^2}{\sqrt {a x^4-b}}+\sqrt {1-\sqrt {4 a b+1}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {1-\sqrt {4 a b+1}}}\right )}{\sqrt {4 a b+1}}-\frac {2 a \left (\frac {\int \frac {1}{\sqrt {\sqrt {4 a b+1}+1}-\frac {\sqrt {a} \sqrt {-2 b+\sqrt {4 a b+1}+1} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {4 a b+1}+1}}+\frac {\int \frac {1}{\frac {\sqrt {a} \sqrt {-2 b+\sqrt {4 a b+1}+1} x^2}{\sqrt {a x^4-b}}+\sqrt {\sqrt {4 a b+1}+1}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {4 a b+1}+1}}\right )}{\sqrt {4 a b+1}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 a \left (\frac {\int \frac {1}{\sqrt {1-\sqrt {4 a b+1}}-\frac {\sqrt {a} \sqrt {-2 b-\sqrt {4 a b+1}+1} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {1-\sqrt {4 a b+1}}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}\right )}{\sqrt {4 a b+1}}-\frac {2 a \left (\frac {\int \frac {1}{\sqrt {\sqrt {4 a b+1}+1}-\frac {\sqrt {a} \sqrt {-2 b+\sqrt {4 a b+1}+1} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {4 a b+1}+1}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}\right )}{\sqrt {4 a b+1}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{1-\sqrt {4 a b+1}} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (1-\sqrt {4 a b+1}\right )^{3/4} \sqrt [4]{-\sqrt {4 a b+1}-2 b+1}}\right )}{\sqrt {4 a b+1}}-\frac {2 a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}{\sqrt [4]{\sqrt {4 a b+1}+1} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {4 a b+1}+1\right )^{3/4} \sqrt [4]{\sqrt {4 a b+1}-2 b+1}}\right )}{\sqrt {4 a b+1}}\) |
(2*a*(ArcTan[(a^(1/4)*(1 - 2*b - Sqrt[1 + 4*a*b])^(1/4)*x)/((1 - Sqrt[1 + 4*a*b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*a^(1/4)*(1 - Sqrt[1 + 4*a*b])^(3/4)* (1 - 2*b - Sqrt[1 + 4*a*b])^(1/4)) + ArcTanh[(a^(1/4)*(1 - 2*b - Sqrt[1 + 4*a*b])^(1/4)*x)/((1 - Sqrt[1 + 4*a*b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*a^(1 /4)*(1 - Sqrt[1 + 4*a*b])^(3/4)*(1 - 2*b - Sqrt[1 + 4*a*b])^(1/4))))/Sqrt[ 1 + 4*a*b] - (2*a*(ArcTan[(a^(1/4)*(1 - 2*b + Sqrt[1 + 4*a*b])^(1/4)*x)/(( 1 + Sqrt[1 + 4*a*b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*a^(1/4)*(1 + Sqrt[1 + 4 *a*b])^(3/4)*(1 - 2*b + Sqrt[1 + 4*a*b])^(1/4)) + ArcTanh[(a^(1/4)*(1 - 2* b + Sqrt[1 + 4*a*b])^(1/4)*x)/((1 + Sqrt[1 + 4*a*b])^(1/4)*(-b + a*x^4)^(1 /4))]/(2*a^(1/4)*(1 + Sqrt[1 + 4*a*b])^(3/4)*(1 - 2*b + Sqrt[1 + 4*a*b])^( 1/4))))/Sqrt[1 + 4*a*b]
3.18.24.3.1 Defintions of rubi rules used
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[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_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r) Int[(d + e*x ^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r) Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (-2 a -1\right ) \textit {\_Z}^{4}+a^{2}-a b +a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-2 a -1\right )}}{4 b}\) | \(75\) |
1/4*sum(1/_R*(_R^4-a)*ln((-_R*x+(a*x^4-b)^(1/4))/x)/(2*_R^4-2*a-1),_R=Root Of(_Z^8+(-2*a-1)*_Z^4+a^2-a*b+a))/b
Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 7.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b - x^{4}\right )}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 1.73 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} - x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-x^4+a x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-a\,x^8+x^4+b\right )} \,d x \]