Integrand size = 49, antiderivative size = 108 \[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\frac {(-b-2 c k) \arctan \left (\frac {(-1+k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{4 (-1+k) k}+\frac {(b-2 c k) \arctan \left (\frac {(1+k) x}{1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 k (1+k)} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.71 (sec) , antiderivative size = 397, normalized size of antiderivative = 3.68, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {1976, 6857, 1117, 1224, 1712, 209} \[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=-\frac {\arctan \left (\frac {(1-k) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right ) (b+2 c k)}{4 (1-k) k}+\frac {\arctan \left (\frac {(k+1) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right ) (b-2 c k)}{4 k (k+1)}+\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} (b-2 c k) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{3/2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} (b+2 c k) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{8 k^{3/2} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {c \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}} \]
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Rule 209
Rule 1117
Rule 1224
Rule 1712
Rule 1976
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {c+b x^2+c k^2 x^4}{\left (-1+k^2 x^4\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {c}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2 c+b x^2}{\left (-1+k^2 x^4\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = c \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\int \frac {2 c+b x^2}{\left (-1+k^2 x^4\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {c \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\int \left (-\frac {b+2 c k}{2 k \left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {b-2 c k}{2 k \left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \frac {c \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {(b-2 c k) \int \frac {1}{\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{2 k}-\frac {(b+2 c k) \int \frac {1}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{2 k} \\ & = \frac {c \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {(b-2 c k) \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{4 k}+\frac {(b-2 c k) \int \frac {1-k x^2}{\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{4 k}-\frac {(b+2 c k) \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{4 k}-\frac {(b+2 c k) \int \frac {1+k x^2}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{4 k} \\ & = \frac {c \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {(b-2 c k) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{8 k^{3/2} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {(b+2 c k) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{8 k^{3/2} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {(b-2 c k) \text {Subst}\left (\int \frac {1}{1-\left (-1-2 k-k^2\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{4 k}-\frac {(b+2 c k) \text {Subst}\left (\int \frac {1}{1-\left (-1+2 k-k^2\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{4 k} \\ & = -\frac {(b+2 c k) \arctan \left (\frac {(1-k) x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{4 (1-k) k}+\frac {(b-2 c k) \arctan \left (\frac {(1+k) x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{4 k (1+k)}+\frac {c \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {(b-2 c k) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{8 k^{3/2} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {(b+2 c k) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{8 k^{3/2} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 11.46 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (2 c k \operatorname {EllipticF}\left (\arcsin (x),k^2\right )+(b-2 c k) \operatorname {EllipticPi}\left (-k,\arcsin (x),k^2\right )-(b+2 c k) \operatorname {EllipticPi}\left (k,\arcsin (x),k^2\right )\right )}{2 k \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
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Time = 2.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99
method | result | size |
elliptic | \(\frac {\left (\frac {\left (2 c k -b \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (1+k \right )}\right )}{4 k \left (1+k \right )}+\frac {\left (2 c k +b \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (-1+k \right )}\right )}{4 k \left (-1+k \right )}\right ) \sqrt {2}}{2}\) | \(107\) |
default | \(-\frac {-2 \left (-2 c k +b \right ) \left (\ln \left (\frac {\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-x \left (1+k \right )^{2}}{k \,x^{2}+1}\right )+\ln \left (2\right )\right ) \sqrt {-\left (-1+k \right )^{2}}+\left (2 c k +b \right ) \left (\ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )+\ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {-\left (1+k \right )^{2}}}{8 \sqrt {-\left (1+k \right )^{2}}\, \sqrt {-\left (-1+k \right )^{2}}\, k}\) | \(253\) |
pseudoelliptic | \(-\frac {-2 \left (-2 c k +b \right ) \left (\ln \left (\frac {\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-x \left (1+k \right )^{2}}{k \,x^{2}+1}\right )+\ln \left (2\right )\right ) \sqrt {-\left (-1+k \right )^{2}}+\left (2 c k +b \right ) \left (\ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )+\ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )+2 \ln \left (2\right )\right ) \sqrt {-\left (1+k \right )^{2}}}{8 \sqrt {-\left (1+k \right )^{2}}\, \sqrt {-\left (-1+k \right )^{2}}\, k}\) | \(253\) |
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Time = 0.61 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\frac {{\left (2 \, c k^{2} - {\left (b + 2 \, c\right )} k + b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}{{\left (k + 1\right )} x}\right ) + {\left (2 \, c k^{2} + {\left (b + 2 \, c\right )} k + b\right )} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}{{\left (k - 1\right )} x}\right )}{4 \, {\left (k^{3} - k\right )}} \]
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\[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\int \frac {b x^{2} + c k^{2} x^{4} + c}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x^{2} - 1\right ) \left (k x^{2} + 1\right )}\, dx \]
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\[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\int { \frac {c k^{2} x^{4} + b x^{2} + c}{{\left (k^{2} x^{4} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
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\[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\int { \frac {c k^{2} x^{4} + b x^{2} + c}{{\left (k^{2} x^{4} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
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Timed out. \[ \int \frac {c+b x^2+c k^2 x^4}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\int \frac {c\,k^2\,x^4+b\,x^2+c}{\left (k^2\,x^4-1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x \]
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