Integrand size = 19, antiderivative size = 49 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {1}{8} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 \sqrt [4]{x^2+x^4}} \]
(Sqrt[x]*(1 + x^2)^(1/4)*RootSum[1 - 4*#1^4 + 2*#1^8 & , (-Log[Sqrt[x]] + Log[(1 + x^2)^(1/4) - Sqrt[x]*#1])/#1 & ])/(8*(x^2 + x^4)^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(244\) vs. \(2(49)=98\).
Time = 0.42 (sec) , antiderivative size = 244, normalized size of antiderivative = 4.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2467, 25, 1593, 1759, 902, 756, 216, 219}
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.
| \(\displaystyle \int \frac {1}{\left (x^4-2\right ) \sqrt [4]{x^4+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{x^2+1} \int -\frac {1}{\sqrt {x} \sqrt [4]{x^2+1} \left (2-x^4\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt [4]{x^2+1} \int \frac {1}{\sqrt {x} \sqrt [4]{x^2+1} \left (2-x^4\right )}dx}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1593 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \int \frac {1}{\sqrt [4]{x^2+1} \left (2-x^4\right )}d\sqrt {x}}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 1759 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {\int \frac {1}{\left (\sqrt {2}-x^2\right ) \sqrt [4]{x^2+1}}d\sqrt {x}}{2 \sqrt {2}}+\frac {\int \frac {1}{\sqrt [4]{x^2+1} \left (x^2+\sqrt {2}\right )}d\sqrt {x}}{2 \sqrt {2}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {\int \frac {1}{\sqrt {2}-\left (-1+\sqrt {2}\right ) x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {2}}+\frac {\int \frac {1}{\sqrt {2}-\left (1+\sqrt {2}\right ) x^2}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {2}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \int \frac {1}{\sqrt {2-\sqrt {2}}-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \int \frac {1}{x+\sqrt {2-\sqrt {2}}}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {2}}+\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \int \frac {1}{\sqrt {2+\sqrt {2}}-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \int \frac {1}{x+\sqrt {2+\sqrt {2}}}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}}{2 \sqrt {2}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \int \frac {1}{\sqrt {2-\sqrt {2}}-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {\sqrt [4]{2-\sqrt {2}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{2 \sqrt {2}}}{2 \sqrt {2}}+\frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \int \frac {1}{\sqrt {2+\sqrt {2}}-x}d\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}+\frac {\sqrt [4]{2+\sqrt {2}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{2 \sqrt {2}}}{2 \sqrt {2}}\right )}{\sqrt [4]{x^4+x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{x^2+1} \left (\frac {\frac {\sqrt [4]{2-\sqrt {2}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{2 \sqrt {2}}+\frac {\sqrt [4]{2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{2 \sqrt {2}}}{2 \sqrt {2}}+\frac {\frac {\sqrt [4]{2+\sqrt {2}} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{2 \sqrt {2}}+\frac {\sqrt [4]{2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{2 \sqrt {2}}}{2 \sqrt {2}}\right )}{\sqrt [4]{x^4+x^2}}\) |
(-2*Sqrt[x]*(1 + x^2)^(1/4)*((((2 - Sqrt[2])^(1/4)*ArcTan[Sqrt[x]/((2 - Sq rt[2])^(1/4)*(1 + x^2)^(1/4))])/(2*Sqrt[2]) + ((2 - Sqrt[2])^(1/4)*ArcTanh [Sqrt[x]/((2 - Sqrt[2])^(1/4)*(1 + x^2)^(1/4))])/(2*Sqrt[2]))/(2*Sqrt[2]) + (((2 + Sqrt[2])^(1/4)*ArcTan[Sqrt[x]/((2 + Sqrt[2])^(1/4)*(1 + x^2)^(1/4 ))])/(2*Sqrt[2]) + ((2 + Sqrt[2])^(1/4)*ArcTanh[Sqrt[x]/((2 + Sqrt[2])^(1/ 4)*(1 + x^2)^(1/4))])/(2*Sqrt[2]))/(2*Sqrt[2])))/(x^2 + x^4)^(1/4)
3.7.24.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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_) + (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))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)], x] ] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> W ith[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r) Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Simp[c/(2*r) Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + 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 = 50.78 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(44\) |
| trager | \(\text {Expression too large to display}\) | \(3668\) |
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.01 (sec) , antiderivative size = 1477, normalized size of antiderivative = 30.14 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \]
-1/16*sqrt(-sqrt(-sqrt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 + sqrt(2) *(6*x^2 + 11) + 12) - 2*(34*x^4 + 46*x^2 + sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(-sqrt(2) + 2) + (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(3 4*x^3 + sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(-sqrt(2) + 2) + sqrt(2)*(35*x ^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(-sqrt(2) + 2)))/(x^5 - 2*x)) + 1/16 *sqrt(-sqrt(-sqrt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 + sqrt(2)*(6*x ^2 + 11) + 12) - 2*(34*x^4 + 46*x^2 + sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^ 2)^(1/4)*sqrt(-sqrt(2) + 2) - (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(34*x^3 + sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(-sqrt(2) + 2) + sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(-sqrt(2) + 2)))/(x^5 - 2*x)) - 1/16*sqrt (-sqrt(sqrt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 - sqrt(2)*(6*x^2 + 1 1) + 12) - 2*(34*x^4 + 46*x^2 - sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/ 4)*sqrt(sqrt(2) + 2) + (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(34*x^3 - sqrt (2)*(23*x^3 + 34*x) + 46*x)*sqrt(sqrt(2) + 2) - sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(sqrt(2) + 2)))/(x^5 - 2*x)) + 1/16*sqrt(-sqrt(sq rt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 - sqrt(2)*(6*x^2 + 11) + 12) - 2*(34*x^4 + 46*x^2 - sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(s qrt(2) + 2) - (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(34*x^3 - sqrt(2)*(23*x ^3 + 34*x) + 46*x)*sqrt(sqrt(2) + 2) - sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(sqrt(2) + 2)))/(x^5 - 2*x)) - 1/16*(-sqrt(2) + 2)^(1/4...
Not integrable
Time = 0.97 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - 2\right )}\, dx \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 2\right )}} \,d x } \]
2/21*(4*x^5 + x^3 - 3*x)/((x^(9/2) - 2*sqrt(x))*(x^2 + 1)^(1/4)) + integra te(16/21*(4*x^4 + x^2 - 3)/((x^(17/2) - 4*x^(9/2) + 4*sqrt(x))*(x^2 + 1)^( 1/4)), x)
Not integrable
Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 2\right )}} \,d x } \]
Not integrable
Time = 5.57 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-2\right )} \,d x \]