Integrand size = 40, antiderivative size = 77 \[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\frac {\arctan \left (\frac {(-1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{1-k}+\frac {\arctan \left (\frac {(1+k) x}{\sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}\right )}{-1-k} \]
arctan((-1+k)*x/(x+(-k^2-1)*x^2+k^2*x^3)^(1/2))/(1-k)+arctan((1+k)*x/(x+(- k^2-1)*x^2+k^2*x^3)^(1/2))/(-1-k)
Time = 11.00 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=-\frac {(1+k) \arctan \left (\frac {(-1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )+(-1+k) \arctan \left (\frac {(1+k) x}{\sqrt {(-1+x) x \left (-1+k^2 x\right )}}\right )}{-1+k^2} \]
-(((1 + k)*ArcTan[((-1 + k)*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]] + (-1 + k)*A rcTan[((1 + k)*x)/Sqrt[(-1 + x)*x*(-1 + k^2*x)]])/(-1 + k^2))
Time = 1.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.71, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
\(\displaystyle \int \frac {k^2 x^2+1}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (k^2 x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int -\frac {k^2 x^2+1}{\sqrt {x} \left (1-k^2 x^2\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {k^2 x^2+1}{\sqrt {x} \left (1-k^2 x^2\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \frac {k^2 x^2+1}{\left (1-k^2 x^2\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \int \left (\frac {2}{\left (1-k^2 x^2\right ) \sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}-\frac {1}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )d\sqrt {x}}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {k^2 x^2-\left (k^2+1\right ) x+1} \left (\frac {\arctan \left (\frac {(1-k) \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{2 (1-k)}+\frac {\arctan \left (\frac {(k+1) \sqrt {x}}{\sqrt {k^2 x^2-\left (k^2+1\right ) x+1}}\right )}{2 (k+1)}\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\) |
(-2*Sqrt[x]*Sqrt[1 - (1 + k^2)*x + k^2*x^2]*(ArcTan[((1 - k)*Sqrt[x])/Sqrt [1 - (1 + k^2)*x + k^2*x^2]]/(2*(1 - k)) + ArcTan[((1 + k)*Sqrt[x])/Sqrt[1 - (1 + k^2)*x + k^2*x^2]]/(2*(1 + k))))/Sqrt[(1 - x)*x*(1 - k^2*x)]
3.11.10.3.1 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 = 1.74 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\left (1+k \right ) \arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )+\arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right ) \left (-1+k \right )}{k^{2}-1}\) | \(66\) |
pseudoelliptic | \(\frac {\left (1+k \right ) \arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (-1+k \right ) x}\right )+\arctan \left (\frac {\sqrt {\left (x -1\right ) x \left (k^{2} x -1\right )}}{\left (1+k \right ) x}\right ) \left (-1+k \right )}{k^{2}-1}\) | \(66\) |
elliptic | \(-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticF}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}-\frac {1}{k}\right )}+\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {\frac {x}{\frac {1}{k^{2}}-1}-\frac {1}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \operatorname {EllipticPi}\left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{k^{3} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k}\right )}\) | \(335\) |
((1+k)*arctan(((x-1)*x*(k^2*x-1))^(1/2)/(-1+k)/x)+arctan(((x-1)*x*(k^2*x-1 ))^(1/2)/(1+k)/x)*(-1+k))/(k^2-1)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (69) = 138\).
Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.26 \[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\frac {{\left (k - 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} + k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} + k^{2}\right )} x^{3} - {\left (k^{3} + k^{2} + k + 1\right )} x^{2} + {\left (k + 1\right )} x\right )}}\right ) + {\left (k + 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, {\left (k^{2} - k + 1\right )} x + 1\right )}}{2 \, {\left ({\left (k^{3} - k^{2}\right )} x^{3} - {\left (k^{3} - k^{2} + k - 1\right )} x^{2} + {\left (k - 1\right )} x\right )}}\right )}{2 \, {\left (k^{2} - 1\right )}} \]
1/2*((k - 1)*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 + 1)*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^2 - 1)
\[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\int \frac {k^{2} x^{2} + 1}{\sqrt {x \left (x - 1\right ) \left (k^{2} x - 1\right )} \left (k x - 1\right ) \left (k x + 1\right )}\, dx \]
\[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\int { \frac {k^{2} x^{2} + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
\[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\int { \frac {k^{2} x^{2} + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}} \,d x } \]
Timed out. \[ \int \frac {1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+k^2 x^2\right )} \, dx=\text {Hanged} \]