Integrand size = 39, antiderivative size = 143 \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=-\frac {\text {RootSum}\left [a^4 c-2 a^4 b^4 \text {$\#$1}^4-c \text {$\#$1}^4+b^4 \text {$\#$1}^8\&,\frac {a^4 \log (x)-a^4 \log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{2 a^4 b^4 \text {$\#$1}+c \text {$\#$1}-2 b^4 \text {$\#$1}^5}\&\right ]}{4 b^4} \]
Time = 0.73 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=-\frac {\text {RootSum}\left [a^4 c-2 a^4 b^4 \text {$\#$1}^4-c \text {$\#$1}^4+b^4 \text {$\#$1}^8\&,\frac {-a^4 \log (x)+a^4 \log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a^4 b^4 \text {$\#$1}-c \text {$\#$1}+2 b^4 \text {$\#$1}^5}\&\right ]}{4 b^4} \]
-1/4*RootSum[a^4*c - 2*a^4*b^4*#1^4 - c*#1^4 + b^4*#1^8 & , (-(a^4*Log[x]) + a^4*Log[(-b^4 + a^4*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b^4 + a^4* x^4)^(1/4) - x*#1]*#1^4)/(-2*a^4*b^4*#1 - c*#1 + 2*b^4*#1^5) & ]/b^4
Leaf count is larger than twice the leaf count of optimal. \(589\) vs. \(2(143)=286\).
Time = 0.99 (sec) , antiderivative size = 589, normalized size of antiderivative = 4.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{a^4 x^4-b^4} \left (a^8 x^8-b^8-c x^4\right )} \, dx\) |
\(\Big \downarrow \) 1758 |
\(\displaystyle \frac {2 a^8 \int -\frac {1}{\sqrt [4]{a^4 x^4-b^4} \left (-2 x^4 a^8+c+\sqrt {4 a^8 b^8+c^2}\right )}dx}{\sqrt {4 a^8 b^8+c^2}}-\frac {2 a^8 \int -\frac {1}{\sqrt [4]{a^4 x^4-b^4} \left (-2 x^4 a^8+c-\sqrt {4 a^8 b^8+c^2}\right )}dx}{\sqrt {4 a^8 b^8+c^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 a^8 \int \frac {1}{\sqrt [4]{a^4 x^4-b^4} \left (-2 x^4 a^8+c-\sqrt {4 a^8 b^8+c^2}\right )}dx}{\sqrt {4 a^8 b^8+c^2}}-\frac {2 a^8 \int \frac {1}{\sqrt [4]{a^4 x^4-b^4} \left (-2 x^4 a^8+c+\sqrt {4 a^8 b^8+c^2}\right )}dx}{\sqrt {4 a^8 b^8+c^2}}\) |
\(\Big \downarrow \) 902 |
\(\displaystyle \frac {2 a^8 \int \frac {1}{-\frac {\left (a^4 \left (c-\sqrt {4 a^8 b^8+c^2}\right )-2 a^8 b^4\right ) x^4}{a^4 x^4-b^4}+c-\sqrt {4 a^8 b^8+c^2}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{\sqrt {4 a^8 b^8+c^2}}-\frac {2 a^8 \int \frac {1}{-\frac {\left (a^4 \left (c+\sqrt {4 a^8 b^8+c^2}\right )-2 a^8 b^4\right ) x^4}{a^4 x^4-b^4}+c+\sqrt {4 a^8 b^8+c^2}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{\sqrt {4 a^8 b^8+c^2}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {2 a^8 \left (-\frac {\int \frac {1}{\sqrt {\sqrt {4 a^8 b^8+c^2}-c}-\frac {a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2}{\sqrt {a^4 x^4-b^4}}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{2 \sqrt {\sqrt {4 a^8 b^8+c^2}-c}}-\frac {\int \frac {1}{\frac {a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2}{\sqrt {a^4 x^4-b^4}}+\sqrt {\sqrt {4 a^8 b^8+c^2}-c}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{2 \sqrt {\sqrt {4 a^8 b^8+c^2}-c}}\right )}{\sqrt {4 a^8 b^8+c^2}}-\frac {2 a^8 \left (\frac {\int \frac {1}{\sqrt {c+\sqrt {4 a^8 b^8+c^2}}-\frac {a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2}{\sqrt {a^4 x^4-b^4}}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{2 \sqrt {\sqrt {4 a^8 b^8+c^2}+c}}+\frac {\int \frac {1}{\frac {a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2}{\sqrt {a^4 x^4-b^4}}+\sqrt {c+\sqrt {4 a^8 b^8+c^2}}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{2 \sqrt {\sqrt {4 a^8 b^8+c^2}+c}}\right )}{\sqrt {4 a^8 b^8+c^2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 a^8 \left (-\frac {\int \frac {1}{\sqrt {\sqrt {4 a^8 b^8+c^2}-c}-\frac {a^2 \sqrt {2 a^4 b^4-c+\sqrt {4 a^8 b^8+c^2}} x^2}{\sqrt {a^4 x^4-b^4}}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{2 \sqrt {\sqrt {4 a^8 b^8+c^2}-c}}-\frac {\arctan \left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{2 a \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}\right )}{\sqrt {4 a^8 b^8+c^2}}-\frac {2 a^8 \left (\frac {\int \frac {1}{\sqrt {c+\sqrt {4 a^8 b^8+c^2}}-\frac {a^2 \sqrt {-2 a^4 b^4+c+\sqrt {4 a^8 b^8+c^2}} x^2}{\sqrt {a^4 x^4-b^4}}}d\frac {x}{\sqrt [4]{a^4 x^4-b^4}}}{2 \sqrt {\sqrt {4 a^8 b^8+c^2}+c}}+\frac {\arctan \left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{2 a \left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}\right )}{\sqrt {4 a^8 b^8+c^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 a^8 \left (-\frac {\arctan \left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{2 a \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}-\frac {\text {arctanh}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}-c} \sqrt [4]{a^4 x^4-b^4}}\right )}{2 a \left (\sqrt {4 a^8 b^8+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}+2 a^4 b^4-c}}\right )}{\sqrt {4 a^8 b^8+c^2}}-\frac {2 a^8 \left (\frac {\arctan \left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{2 a \left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}+\frac {\text {arctanh}\left (\frac {a x \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}{\sqrt [4]{\sqrt {4 a^8 b^8+c^2}+c} \sqrt [4]{a^4 x^4-b^4}}\right )}{2 a \left (\sqrt {4 a^8 b^8+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {4 a^8 b^8+c^2}-2 a^4 b^4+c}}\right )}{\sqrt {4 a^8 b^8+c^2}}\) |
(2*a^8*(-1/2*ArcTan[(a*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/(( -c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4))]/(a*(-c + Sqrt[4 *a^8*b^8 + c^2])^(3/4)*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)) - Ar cTanh[(a*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((-c + Sqrt[4*a^ 8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4))]/(2*a*(-c + Sqrt[4*a^8*b^8 + c ^2])^(3/4)*(2*a^4*b^4 - c + Sqrt[4*a^8*b^8 + c^2])^(1/4))))/Sqrt[4*a^8*b^8 + c^2] - (2*a^8*(ArcTan[(a*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4) *x)/((c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4))]/(2*a*(c + Sqrt[4*a^8*b^8 + c^2])^(3/4)*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4 )) + ArcTanh[(a*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4)*x)/((c + Sq rt[4*a^8*b^8 + c^2])^(1/4)*(-b^4 + a^4*x^4)^(1/4))]/(2*a*(c + Sqrt[4*a^8*b ^8 + c^2])^(3/4)*(-2*a^4*b^4 + c + Sqrt[4*a^8*b^8 + c^2])^(1/4))))/Sqrt[4* a^8*b^8 + c^2]
3.21.20.3.1 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 = 0.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} b^{4}+\left (-2 a^{4} b^{4}-c \right ) \textit {\_Z}^{4}+a^{4} c \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+a^{4}\right ) \ln \left (\frac {-\textit {\_R} x +\left (a^{4} x^{4}-b^{4}\right )^{\frac {1}{4}}}{x}\right )}{-2 b^{4} \textit {\_R}^{5}+\left (2 a^{4} b^{4}+c \right ) \textit {\_R}}}{4 b^{4}}\) | \(97\) |
1/4*sum((-_R^4+a^4)*ln((-_R*x+(a^4*x^4-b^4)^(1/4))/x)/(-2*b^4*_R^5+(2*a^4* b^4+c)*_R),_R=RootOf(_Z^8*b^4+(-2*a^4*b^4-c)*_Z^4+a^4*c))/b^4
Timed out. \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 10.63 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a^{8} x^{8} - b^{8} - c x^{4}\right )}\, dx \]
Integral(1/(((a*x - b)*(a*x + b)*(a**2*x**2 + b**2))**(1/4)*(a**8*x**8 - b **8 - c*x**4)), x)
Not integrable
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=\int { \frac {1}{{\left (a^{8} x^{8} - b^{8} - c x^{4}\right )} {\left (a^{4} x^{4} - b^{4}\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 2.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=\int { \frac {1}{{\left (a^{8} x^{8} - b^{8} - c x^{4}\right )} {\left (a^{4} x^{4} - b^{4}\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 6.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b^4+a^4 x^4} \left (-b^8-c x^4+a^8 x^8\right )} \, dx=-\int \frac {1}{{\left (a^4\,x^4-b^4\right )}^{1/4}\,\left (-a^8\,x^8+b^8+c\,x^4\right )} \,d x \]