Integrand size = 29, antiderivative size = 106 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=-\frac {\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{4 b} \]
Time = 0.49 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=-\frac {\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{4 b} \]
-1/4*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (-(a*Log[x]) + a*Log[(-b + a* x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b + a*x^4)^(1/4) - x*#1]*#1^4)/(- 3*a*#1 + 2*#1^5) & ]/b
Leaf count is larger than twice the leaf count of optimal. \(449\) vs. \(2(106)=212\).
Time = 0.74 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{a x^4-b} \left (-a x^4-b+x^8\right )} \, dx\) |
\(\Big \downarrow \) 1758 |
\(\displaystyle \frac {2 \int -\frac {1}{\left (-2 x^4+a+\sqrt {a^2+4 b}\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {a^2+4 b}}-\frac {2 \int -\frac {1}{\left (-2 x^4+a-\sqrt {a^2+4 b}\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {a^2+4 b}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \int \frac {1}{\left (-2 x^4+a-\sqrt {a^2+4 b}\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {a^2+4 b}}-\frac {2 \int \frac {1}{\left (-2 x^4+a+\sqrt {a^2+4 b}\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt {a^2+4 b}}\) |
\(\Big \downarrow \) 902 |
\(\displaystyle \frac {2 \int \frac {1}{-\frac {\left (a \left (a-\sqrt {a^2+4 b}\right )-2 b\right ) x^4}{a x^4-b}+a-\sqrt {a^2+4 b}}d\frac {x}{\sqrt [4]{a x^4-b}}}{\sqrt {a^2+4 b}}-\frac {2 \int \frac {1}{-\frac {\left (a \left (a+\sqrt {a^2+4 b}\right )-2 b\right ) x^4}{a x^4-b}+a+\sqrt {a^2+4 b}}d\frac {x}{\sqrt [4]{a x^4-b}}}{\sqrt {a^2+4 b}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\frac {\sqrt {a^2-\sqrt {a^2+4 b} a-2 b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a-\sqrt {a^2+4 b}}}+\frac {\int \frac {1}{\frac {\sqrt {a^2-\sqrt {a^2+4 b} a-2 b} x^2}{\sqrt {a x^4-b}}+\sqrt {a-\sqrt {a^2+4 b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a-\sqrt {a^2+4 b}}}\right )}{\sqrt {a^2+4 b}}-\frac {2 \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\frac {\sqrt {a^2+\sqrt {a^2+4 b} a-2 b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {a^2+4 b}+a}}+\frac {\int \frac {1}{\frac {\sqrt {a^2+\sqrt {a^2+4 b} a-2 b} x^2}{\sqrt {a x^4-b}}+\sqrt {a+\sqrt {a^2+4 b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {a^2+4 b}+a}}\right )}{\sqrt {a^2+4 b}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\frac {\sqrt {a^2-\sqrt {a^2+4 b} a-2 b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {a-\sqrt {a^2+4 b}}}+\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt {a^2+4 b}}-\frac {2 \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\frac {\sqrt {a^2+\sqrt {a^2+4 b} a-2 b} x^2}{\sqrt {a x^4-b}}}d\frac {x}{\sqrt [4]{a x^4-b}}}{2 \sqrt {\sqrt {a^2+4 b}+a}}+\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt {a^2+4 b}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \left (\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt {a^2+4 b}}-\frac {2 \left (\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2+4 b}+a} \sqrt [4]{a x^4-b}}\right )}{2 \left (\sqrt {a^2+4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+4 b}+a^2-2 b}}\right )}{\sqrt {a^2+4 b}}\) |
(2*(ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a - Sqrt[a^2 + 4*b] )^(1/4)*(-b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a *Sqrt[a^2 + 4*b])^(1/4)) + ArcTanh[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)* x)/((a - Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 + 4* b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4))))/Sqrt[a^2 + 4*b] - (2*(A rcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x)/((a + Sqrt[a^2 + 4*b])^(1/ 4)*(-b + a*x^4)^(1/4))]/(2*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt [a^2 + 4*b])^(1/4)) + ArcTanh[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x)/(( a + Sqrt[a^2 + 4*b])^(1/4)*(-b + a*x^4)^(1/4))]/(2*(a + Sqrt[a^2 + 4*b])^( 3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4))))/Sqrt[a^2 + 4*b]
3.16.46.3.1 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 = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-3 a \right )}}{4 b}\) | \(71\) |
1/4*sum(1/_R*(_R^4-a)*ln((-_R*x+(a*x^4-b)^(1/4))/x)/(2*_R^4-3*a),_R=RootOf (_Z^8-3*_Z^4*a+2*a^2-b))/b
Timed out. \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 7.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} - b} \left (- a x^{4} - b + x^{8}\right )}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 1.55 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 5.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b-a x^4+x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-x^8+a\,x^4+b\right )} \,d x \]