Integrand size = 46, antiderivative size = 178 \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\frac {\left (-b-2 a k^2\right ) \arctan \left (\frac {(-1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{4 (-1+k) k^2}+\frac {\left (-b-2 a k^2\right ) \arctan \left (\frac {(1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{4 k^2 (1+k)}+\frac {\left (b-2 a k^2\right ) \arctan \left (\frac {\sqrt {1+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{(-1+x) \left (-1+k^2 x\right )}\right )}{2 k^2 \sqrt {1+k^2}} \]
1/4*(-2*a*k^2-b)*arctan((-1+k)*x/(x+(-k^2-1)*x^2+k^2*x^3)^(1/2))/(-1+k)/k^ 2+1/4*(-2*a*k^2-b)*arctan((1+k)*x/(x+(-k^2-1)*x^2+k^2*x^3)^(1/2))/k^2/(1+k )+1/2*(-2*a*k^2+b)*arctan((k^2+1)^(1/2)*(x+(-k^2-1)*x^2+k^2*x^3)^(1/2)/(-1 +x)/(k^2*x-1))/k^2/(k^2+1)^(1/2)
Time = 12.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=-\frac {\frac {\left (b+2 a k^2\right ) \arctan \left (\frac {(-1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{-1+k}+\frac {\left (b+2 a k^2\right ) \arctan \left (\frac {(1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{1+k}-\frac {2 \left (b-2 a k^2\right ) \arctan \left (\frac {\sqrt {1+k^2} x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{\sqrt {1+k^2}}}{4 k^2} \]
-1/4*(((b + 2*a*k^2)*ArcTan[((-1 + k)*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]])/( -1 + k) + ((b + 2*a*k^2)*ArcTan[((1 + k)*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]] )/(1 + k) - (2*(b - 2*a*k^2)*ArcTan[(Sqrt[1 + k^2]*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]])/Sqrt[1 + k^2])/k^2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.36 (sec) , antiderivative size = 639, normalized size of antiderivative = 3.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {2467, 25, 2035, 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.
| \(\displaystyle \int \frac {a k^4 x^4+a+b x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^4 x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {a k^4 x^4+b x^2+a}{\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (1-k^4 x^4\right )}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {a k^4 x^4+b x^2+a}{\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (1-k^4 x^4\right )}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {a k^4 x^4+b x^2+a}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (1-k^4 x^4\right )}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {b x^2+2 a}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (1-k^4 x^4\right )}-\frac {a}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (\frac {\left (2 a k^2+b\right ) \arctan \left (\frac {(1-k) \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{8 (1-k) k^2}+\frac {\left (2 a k^2+b\right ) \arctan \left (\frac {(k+1) \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{8 k^2 (k+1)}+\frac {\left (b-2 a k^2\right ) \arctan \left (\frac {\sqrt {k^2+1} \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{4 \sqrt {-k^2} \sqrt {-k^2 \left (k^2+1\right )}}+\frac {(k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \left (2 a k^2+b\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{5/2} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (k-\sqrt {-k^2}\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \left (b-2 a k^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{16 k^{7/2} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {\left (\sqrt {-k^2}+k\right ) (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \left (b-2 a k^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{16 k^{7/2} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {a (k x+1) \sqrt {\frac {k^2 x^2-\left (k^2+1\right ) x+1}{(k x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} \sqrt {x}\right ),\frac {(k+1)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
(-2*Sqrt[x]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]*(((b + 2*a*k^2)*ArcTan[((1 - k )*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]])/(8*(1 - k)*k^2) + ((b + 2*a*k ^2)*ArcTan[((1 + k)*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]])/(8*k^2*(1 + k)) + ((b - 2*a*k^2)*ArcTan[(Sqrt[1 + k^2]*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]])/(4*Sqrt[-k^2]*Sqrt[-(k^2*(1 + k^2))]) - (a*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(2*Sqrt[k]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) + ((b + 2*a *k^2)*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2* ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(8*k^(5/2)*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) - ((b - 2*a*k^2)*(k - Sqrt[-k^2])*(1 + k*x)*Sqrt[(1 - (1 + k^ 2)*x + k^2*x^2)/(1 + k*x)^2]*EllipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^ 2/(4*k)])/(16*k^(7/2)*Sqrt[1 - (1 + k^2)*x + k^2*x^2]) - ((b - 2*a*k^2)*(k + Sqrt[-k^2])*(1 + k*x)*Sqrt[(1 - (1 + k^2)*x + k^2*x^2)/(1 + k*x)^2]*Ell ipticF[2*ArcTan[Sqrt[k]*Sqrt[x]], (1 + k)^2/(4*k)])/(16*k^(7/2)*Sqrt[1 - ( 1 + k^2)*x + k^2*x^2])))/Sqrt[(1 - x)*x*(1 - k^2*x)]
3.24.14.3.1 Defintions of rubi rules used
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 = 1.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\frac {\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right )}{2+2 k}+\frac {\left (2 a \,k^{2}-b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {k^{2}+1}}\right )}{\sqrt {k^{2}+1}}+\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )}{-2+2 k}}{2 k^{2}}\) | \(129\) |
| pseudoelliptic | \(\frac {\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right )}{2+2 k}+\frac {\left (2 a \,k^{2}-b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{x \sqrt {k^{2}+1}}\right )}{\sqrt {k^{2}+1}}+\frac {\left (2 a \,k^{2}+b \right ) \arctan \left (\frac {\sqrt {\left (-1+x \right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )}{-2+2 k}}{2 k^{2}}\) | \(129\) |
| elliptic | \(-\frac {2 a \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-1\right )}-\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{2 k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-1\right )}-\frac {\left (2 a \,k^{2}-b \right ) \sqrt {-k^{2} x +1}\, \sqrt {-\frac {\left (-1+x \right ) k^{2}}{k^{2}-1}}\, \sqrt {k^{2} x}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2} k^{2}+1\right )}{\sum }\operatorname {EllipticPi}\left (\sqrt {-k^{2} x +1}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{2}+1}, \sqrt {-\frac {1}{k^{2}-1}}\right )\right )}{2 k^{2} \left (k^{2}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}-\frac {\left (2 a \,k^{2}-b \right ) \sqrt {-k^{2} x +1}\, \sqrt {-\frac {\left (-1+x \right ) k^{2}}{k^{2}-1}}\, \sqrt {k^{2} x}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2} k^{2}+1\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {-k^{2} x +1}, \frac {\underline {\hspace {1.25 ex}}\alpha \,k^{2}+1}{k^{2}+1}, \sqrt {-\frac {1}{k^{2}-1}}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2 k^{4} \left (k^{2}+1\right ) \sqrt {x \left (k^{2} x^{2}-k^{2} x -x +1\right )}}+\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) a}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+1\right )}+\frac {\sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+1\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) b}{2 k^{4} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+1\right )}\) | \(813\) |
1/2/k^2*(1/2*(2*a*k^2+b)*arctan(((-1+x)*x*(k^2*x-1))^(1/2)/(1+k)/x)/(1+k)+ (2*a*k^2-b)/(k^2+1)^(1/2)*arctan(((-1+x)*x*(k^2*x-1))^(1/2)/x/(k^2+1)^(1/2 ))+1/2*(2*a*k^2+b)*arctan(((-1+x)*x*(k^2*x-1))^(1/2)/(-1+k)/x)/(-1+k))
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (157) = 314\).
Time = 0.73 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.00 \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\frac {2 \, {\left (2 \, a k^{4} - {\left (2 \, a + b\right )} k^{2} + b\right )} \sqrt {k^{2} + 1} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + 1\right )} x + 1\right )} \sqrt {k^{2} + 1}}{2 \, {\left ({\left (k^{4} + k^{2}\right )} x^{3} - {\left (k^{4} + 2 \, k^{2} + 1\right )} x^{2} + {\left (k^{2} + 1\right )} x\right )}}\right ) + {\left (2 \, a k^{5} - 2 \, a k^{4} + {\left (2 \, a + b\right )} k^{3} - {\left (2 \, a + b\right )} k^{2} + b k - b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right ) + {\left (2 \, a k^{5} + 2 \, a k^{4} + {\left (2 \, a + b\right )} k^{3} + {\left (2 \, a + b\right )} k^{2} + b k + b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} - k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} - k^{2}\right )} x^{3} - {\left (k^{3} - k^{2} + k - 1\right )} x^{2} + {\left (k - 1\right )} x\right )}}\right )}{8 \, {\left (k^{6} - k^{2}\right )}} \]
1/8*(2*(2*a*k^4 - (2*a + b)*k^2 + b)*sqrt(k^2 + 1)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2*(k^2 + 1)*x + 1)*sqrt(k^2 + 1)/((k^4 + k^2)*x^3 - (k^4 + 2*k^2 + 1)*x^2 + (k^2 + 1)*x)) + (2*a*k^5 - 2*a*k^4 + (2 *a + b)*k^3 - (2*a + b)*k^2 + b*k - b)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1) *x^2 + x)*(k^2*x^2 - 2*(k^2 + k + 1)*x + 1)/((k^3 + k^2)*x^3 - (k^3 + k^2 + k + 1)*x^2 + (k + 1)*x)) + (2*a*k^5 + 2*a*k^4 + (2*a + b)*k^3 + (2*a + b )*k^2 + b*k + b)*arctan(1/2*sqrt(k^2*x^3 - (k^2 + 1)*x^2 + x)*(k^2*x^2 - 2 *(k^2 - k + 1)*x + 1)/((k^3 - k^2)*x^3 - (k^3 - k^2 + k - 1)*x^2 + (k - 1) *x)))/(k^6 - k^2)
\[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\int \frac {a k^{4} x^{4} + a + b x^{2}}{\sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k x - 1\right ) \left (k x + 1\right ) \left (k^{2} x^{2} + 1\right )}\, dx \]
Integral((a*k**4*x**4 + a + b*x**2)/(sqrt(x*(x - 1)*(k**2*x - 1))*(k*x - 1 )*(k*x + 1)*(k**2*x**2 + 1)), x)
\[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\int { \frac {a k^{4} x^{4} + b x^{2} + a}{{\left (k^{4} x^{4} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
\[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\int { \frac {a k^{4} x^{4} + b x^{2} + a}{{\left (k^{4} x^{4} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
Timed out. \[ \int \frac {a+b x^2+a k^4 x^4}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^4 x^4\right )} \, dx=\text {Hanged} \]