Integrand size = 30, antiderivative size = 104 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{2 b} \]
Time = 0.00 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-2 a \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{4 b \sqrt [4]{-b x^2+a x^4}} \]
(Sqrt[x]*(-b + a*x^2)^(1/4)*RootSum[b - a*#1^4 + #1^8 & , (a*Log[x] - 2*a* Log[(-b + a*x^2)^(1/4) - Sqrt[x]*#1] - Log[x]*#1^4 + 2*Log[(-b + a*x^2)^(1 /4) - Sqrt[x]*#1]*#1^4)/(a*#1 - 2*#1^5) & ])/(4*b*(-(b*x^2) + a*x^4)^(1/4) )
Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(104)=208\).
Time = 0.67 (sec) , antiderivative size = 501, normalized size of antiderivative = 4.82, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2467, 1592, 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.
\(\displaystyle \int \frac {1}{\left (-a x^2+b+x^4\right ) \sqrt [4]{a x^4-b x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2-b} \int \frac {1}{\sqrt {x} \sqrt [4]{a x^2-b} \left (x^4-a x^2+b\right )}dx}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 1592 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \int \frac {1}{\sqrt [4]{a x^2-b} \left (x^4-a x^2+b\right )}d\sqrt {x}}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 1758 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {2 \int -\frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2-b}}d\sqrt {x}}{\sqrt {a^2-4 b}}-\frac {2 \int -\frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2-b}}d\sqrt {x}}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {2 \int \frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2-b}}d\sqrt {x}}{\sqrt {a^2-4 b}}-\frac {2 \int \frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2-b}}d\sqrt {x}}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 902 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {2 \int \frac {1}{-\left (\left (a \left (a-\sqrt {a^2-4 b}\right )-2 b\right ) x^2\right )+a-\sqrt {a^2-4 b}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{\sqrt {a^2-4 b}}-\frac {2 \int \frac {1}{-\left (\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^2\right )+a+\sqrt {a^2-4 b}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {2 \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}+\frac {\int \frac {1}{\sqrt {a^2-\sqrt {a^2-4 b} a-2 b} x+\sqrt {a-\sqrt {a^2-4 b}}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}+\frac {\int \frac {1}{\sqrt {a^2+\sqrt {a^2-4 b} a-2 b} x+\sqrt {a+\sqrt {a^2-4 b}}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {2 \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2-b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \left (\frac {2 \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4-b x^2}}\) |
(2*Sqrt[x]*(-b + a*x^2)^(1/4)*((2*(ArcTan[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b) ^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4)*(-b + a*x^2)^(1/4))]/(2*(a - Sqrt[a^2 - 4*b])^(3/4)*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)) + ArcTanh[(( a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4) *(-b + a*x^2)^(1/4))]/(2*(a - Sqrt[a^2 - 4*b])^(3/4)*(a^2 - a*Sqrt[a^2 - 4 *b] - 2*b)^(1/4))))/Sqrt[a^2 - 4*b] - (2*(ArcTan[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4)*(-b + a*x^2)^(1/4))]/( 2*(a + Sqrt[a^2 - 4*b])^(3/4)*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)) + Arc Tanh[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b] )^(1/4)*(-b + a*x^2)^(1/4))]/(2*(a + Sqrt[a^2 - 4*b])^(3/4)*(a^2 + a*Sqrt[ a^2 - 4*b] - 2*b)^(1/4))))/Sqrt[a^2 - 4*b]))/(-(b*x^2) + a*x^4)^(1/4)
3.15.81.3.1 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[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[ Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*(x^( 4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x ] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
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]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a \,\textit {\_Z}^{4}+b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-a \right )}}{2 b}\) | \(68\) |
-1/2*sum(1/_R*(_R^4-a)*ln((-_R*x+(x^2*(a*x^2-b))^(1/4))/x)/(2*_R^4-a),_R=R ootOf(_Z^8-_Z^4*a+b))/b
Timed out. \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\text {Timed out} \]
Not integrable
Time = 1.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (- a x^{2} + b + x^{4}\right )}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}} \,d x } \]
Not integrable
Time = 3.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {1}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^4-a\,x^2+b\right )} \,d x \]