Integrand size = 45, antiderivative size = 311 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\arctan \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-a b x+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt {a} \sqrt {b}} \]
-1/6*arctan(2*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x)^(1/2)/(a^2*x^2-2*a*b*x-b^2)) /a^(1/2)/b^(1/2)-1/3*2^(1/2)*arctan(2^(1/2)*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x )^(1/2)/(a^2*x^2-a*b*x-b^2))/a^(1/2)/b^(1/2)-1/6*arctanh((-1/2*b^(3/2)/a^( 1/2)+a^(1/2)*b^(1/2)*x+1/2*a^(3/2)*x^2/b^(1/2))/(a^2*x^3-b^2*x)^(1/2))/a^( 1/2)/b^(1/2)-1/3*2^(1/2)*arctanh((-1/2*b^(3/2)*2^(1/2)/a^(1/2)+1/2*a^(1/2) *b^(1/2)*x*2^(1/2)+1/2*a^(3/2)*x^2*2^(1/2)/b^(1/2))/(a^2*x^3-b^2*x)^(1/2)) /a^(1/2)/b^(1/2)
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.76 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\frac {\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2} \left (i \arctan \left (\frac {(1+i) \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+(2-2 i) (-1)^{3/4} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+(2-2 i) \sqrt [4]{-1} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+\arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^2+a^2 x^2}}{\sqrt {a} \sqrt {b} \sqrt {x}}\right )\right )}{\sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]
((1/6 + I/6)*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*(I*ArcTan[((1 + I)*Sqrt[a]*Sqrt[ b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] + (2 - 2*I)*(-1)^(3/4)*ArcTan[((-1)^(1/4 )*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] + (2 - 2*I)*(-1)^(1/4)*Ar cTan[((-1)^(3/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] + ArcTan[( (1/2 + I/2)*Sqrt[-b^2 + a^2*x^2])/(Sqrt[a]*Sqrt[b]*Sqrt[x])]))/(Sqrt[a]*Sq rt[b]*Sqrt[-(b^2*x) + a^2*x^3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 6.62 (sec) , antiderivative size = 1265, normalized size of antiderivative = 4.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2467, 25, 2019, 2035, 25, 7276, 2009}
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.
| \(\displaystyle \int \frac {a^6 x^6-b^6}{\sqrt {a^2 x^3-b^2 x} \left (a^6 x^6+b^6\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int -\frac {b^6-a^6 x^6}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^6+a^6 x^6\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^6-a^6 x^6}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^6+a^6 x^6\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {\sqrt {a^2 x^2-b^2} \left (-b^4-a^2 x^2 b^2-a^4 x^4\right )}{\sqrt {x} \left (b^6+a^6 x^6\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int -\frac {\sqrt {a^2 x^2-b^2} \left (b^4+a^2 x^2 b^2+a^4 x^4\right )}{b^6+a^6 x^6}d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {\sqrt {a^2 x^2-b^2} \left (b^4+a^2 x^2 b^2+a^4 x^4\right )}{b^6+a^6 x^6}d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \left (\frac {\sqrt {a^2 x^2-b^2} \left (\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}-\sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\sqrt {a^2 x^2-b^2} \left (\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}-i \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\sqrt {a^2 x^2-b^2} \left (\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}+i \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\sqrt {a^2 x^2-b^2} \left (\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right )}{12 b^6 \left (\sqrt {b}+\sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}-\sqrt [6]{-1} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}+\sqrt [6]{-1} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}-\sqrt [3]{-1} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}+\sqrt [3]{-1} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}-(-1)^{2/3} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}+(-1)^{2/3} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}-(-1)^{5/6} \sqrt [12]{-a^6} \sqrt {x}\right )}+\frac {\left (\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+b^{9/2}\right ) \sqrt {a^2 x^2-b^2}}{12 b^6 \left (\sqrt {b}+(-1)^{5/6} \sqrt [12]{-a^6} \sqrt {x}\right )}\right )d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \left (\frac {(-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} \sqrt [3]{-a^6} a^2-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{6 a^{11/2} \sqrt {a^2 x^2-b^2}}+\frac {(-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} \sqrt [3]{-a^6} a^2-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{6 a^{11/2} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left (\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+(-1)^{2/3} a^4+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{6 a^{11/2} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left (\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+(-1)^{2/3} a^4+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{6 a^{11/2} \sqrt {a^2 x^2-b^2}}-\frac {\left (a-\sqrt [6]{-a^6}\right ) \left (a^4+\sqrt [3]{-a^6} a^2+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{6 a^{11/2} \sqrt {a^2 x^2-b^2}}-\frac {\left (a+\sqrt [6]{-a^6}\right ) \left (a^4+\sqrt [3]{-a^6} a^2+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{6 a^{11/2} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {a^5}{\left (-a^6\right )^{5/6}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [6]{-a^6}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}\right )}{\sqrt {a^2 x^3-b^2 x}}\) |
(-2*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*(((-1)^(2/3)*(a - (-1)^(2/3)*(-a^6)^(1/6) )*((-1)^(1/3)*a^4 - (-1)^(2/3)*a^2*(-a^6)^(1/3) - (-a^6)^(2/3))*Sqrt[b]*Sq rt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(6 *a^(11/2)*Sqrt[-b^2 + a^2*x^2]) + ((-1)^(2/3)*(a + (-1)^(2/3)*(-a^6)^(1/6) )*((-1)^(1/3)*a^4 - (-1)^(2/3)*a^2*(-a^6)^(1/3) - (-a^6)^(2/3))*Sqrt[b]*Sq rt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(6 *a^(11/2)*Sqrt[-b^2 + a^2*x^2]) + ((-1)^(1/3)*(a - (-1)^(1/3)*(-a^6)^(1/6) )*((-1)^(2/3)*a^4 + ((-1)^(1/3)*a^8)/(-a^6)^(2/3) + (-a^6)^(2/3))*Sqrt[b]* Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/ (6*a^(11/2)*Sqrt[-b^2 + a^2*x^2]) + ((-1)^(1/3)*(a + (-1)^(1/3)*(-a^6)^(1/ 6))*((-1)^(2/3)*a^4 + ((-1)^(1/3)*a^8)/(-a^6)^(2/3) + (-a^6)^(2/3))*Sqrt[b ]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1] )/(6*a^(11/2)*Sqrt[-b^2 + a^2*x^2]) - ((a - (-a^6)^(1/6))*(a^4 + a^2*(-a^6 )^(1/3) + (-a^6)^(2/3))*Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[( Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(6*a^(11/2)*Sqrt[-b^2 + a^2*x^2]) - ((a + (-a^6)^(1/6))*(a^4 + a^2*(-a^6)^(1/3) + (-a^6)^(2/3))*Sqrt[b]*Sqrt[1 - (a^ 2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(6*a^(11/2)* Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[a^5/(- a^6)^(5/6), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[((-1)^(1/3)*a^...
3.29.82.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
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]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.50 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(-\frac {\left (4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \ln \left (\frac {a^{2} x^{2}-\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}{a^{2} x^{2}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}\right )-4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )+2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )}{12 \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(428\) |
| pseudoelliptic | \(-\frac {\left (4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \ln \left (\frac {a^{2} x^{2}-\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}{a^{2} x^{2}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}\right )-4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )+2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )}{12 \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\) | \(428\) |
| elliptic | \(\text {Expression too large to display}\) | \(847\) |
-1/12*((4*arctan(((a^2*b^2)^(1/4)*x-2^(1/2)*(a^2*x^3-b^2*x)^(1/2))/(a^2*b^ 2)^(1/4)/x)-2*ln((a^2*x^2-(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1/2)+(a ^2*b^2)^(1/2)*x-b^2)/(a^2*x^2+(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1/2 )+(a^2*b^2)^(1/2)*x-b^2))-4*arctan(((a^2*b^2)^(1/4)*x+2^(1/2)*(a^2*x^3-b^2 *x)^(1/2))/(a^2*b^2)^(1/4)/x))*2^(1/2)-2*arctan(((a^2*b^2)^(1/4)*x+(a^2*x^ 3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x)-ln((a^2*x^2+2*(a^2*b^2)^(1/2)*x-b^2-2*( a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2))/(a^2*x^2+2*(a^2*b^2)^(1/4)*(a^2*x^3- b^2*x)^(1/2)+2*(a^2*b^2)^(1/2)*x-b^2))+2*arctan(((a^2*b^2)^(1/4)*x-(a^2*x^ 3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x))/(a^2*b^2)^(1/4)
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 1317, normalized size of antiderivative = 4.23 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\text {Too large to display} \]
1/12*(1/4)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 + 8*((1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) - 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) - 1/12*(1/ 4)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*((1/4 )^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^ 4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) - 4*(a^4*b^2*x^3 - a^2*b^4* x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) - 1/12*I*(1/4)^(1/ 4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(I*(1/4)^(1 /4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(3/4)*(-I*a^4*b^2*x^2 + I*a^2*b ^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 4*(a^4*b^2*x^3 - a^2*b^4 *x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + 1/12*I*(1/4)^(1 /4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(-I*(1/4)^ (1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(3/4)*(I*a^4*b^2*x^2 - I*a^2* b^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 4*(a^4*b^2*x^3 - a^2*b^ 4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + 1/6*(-1/(a^2*b ^2))^(1/4)*log((a^4*x^4 - 3*a^2*b^2*x^2 + b^4 + 2*(a^2*b^2*x*(-1/(a^2*b^2) )^(1/4) + (a^4*b^2*x^2 - a^2*b^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2 *x) - 2*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)))/(a^4*x^4 - a^2*b^2*x ^2 + b^4)) - 1/6*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 3*a^2*b^2*x^2 + b^...
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \]
Integral((a*x - b)*(a*x + b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)/(sqrt(x*(a*x - b)*(a*x + b))*(a**2*x**2 + b**2)*(a**4*x**4 - a**2 *b**2*x**2 + b**4)), x)
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
\[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
Timed out. \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\text {Hanged} \]