Optimal. Leaf size=189 \[ \frac {i \tan ^{-1}\left (\frac {\sqrt {k^2+2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 x-1}\right )}{2 \sqrt {k^2-1} \sqrt {k^2+2 i \sqrt {k^2-1}-2}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {k^2-2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 x-1}\right )}{2 \sqrt {k^2-1} \sqrt {k^2-2 i \sqrt {k^2-1}-2}} \]
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Rubi [C] time = 2.39, antiderivative size = 455, normalized size of antiderivative = 2.41, number of steps used = 26, number of rules used = 14, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.342, Rules used = {1593, 6718, 21, 6688, 6728, 922, 934, 12, 168, 537, 843, 714, 110, 115} \begin {gather*} \frac {\left (-k^2-\sqrt {1-k^2}+1\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{1-\sqrt {1-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (-k^2+\sqrt {1-k^2}+1\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{\sqrt {1-k^2}+1};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (-k^2-\sqrt {1-k^2}+2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (-k^2+\sqrt {1-k^2}+2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 12
Rule 21
Rule 110
Rule 115
Rule 168
Rule 537
Rule 714
Rule 843
Rule 922
Rule 934
Rule 1593
Rule 6688
Rule 6718
Rule 6728
Rubi steps
\begin {align*} \int \frac {-x+x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx &=\int \frac {(-1+x) x}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (1-2 x+k^2 x^2\right )} \, dx\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {(-1+x) \sqrt {x}}{\sqrt {1-x} \sqrt {1-k^2 x} \left (1-2 x+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {1-x} \sqrt {x}}{\sqrt {1-k^2 x} \left (1-2 x+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {x-x^2}}{\sqrt {1-k^2 x} \left (1-2 x+k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (-\frac {k^2 \sqrt {x-x^2}}{\sqrt {1-k^2} \left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}}-\frac {k^2 \sqrt {x-x^2}}{\sqrt {1-k^2} \sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {x-x^2}}{\left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}} \, dx}{\sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {\sqrt {x-x^2}}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{\sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-2+2 k^2-2 \sqrt {1-k^2}-2 k^2 x}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {2-2 k^2-2 \sqrt {1-k^2}+2 k^2 x}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{4 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {2} \left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\sqrt {2} \left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=-\frac {\left (\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{2 k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right )} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \left (2+2 \sqrt {1-k^2}-2 k^2 x\right ) \sqrt {1-k^2 x}} \, dx}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=-\frac {\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-\sqrt {1-k^2}\right ) \left (1-k^2-\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \left (-2 \left (1-\sqrt {1-k^2}\right )-2 k^2 x^2\right ) \sqrt {1+k^2 x^2}} \, dx,x,\sqrt {-x}\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \left (1+\sqrt {1-k^2}\right ) \left (1-k^2+\sqrt {1-k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (2 \left (1+\sqrt {1-k^2}\right )+2 k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=-\frac {\left (2-k^2-\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2-k^2+\sqrt {1-k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{k^2 \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (1-k^2-\sqrt {1-k^2}\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{1-\sqrt {1-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (1-k^2+\sqrt {1-k^2}\right ) (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (\frac {1}{1+\sqrt {1-k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{\left (-k^2\right )^{3/2} \sqrt {1-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ \end {align*}
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Mathematica [C] time = 2.22, size = 227, normalized size = 1.20 \begin {gather*} -\frac {i \sqrt {x-1} \sqrt {x} \sqrt {\frac {k^2 x-1}{k^2-1}} \left (2 \sqrt {1-k^2} F\left (i \sinh ^{-1}\left (\sqrt {x-1}\right )|\frac {k^2}{k^2-1}\right )-\left (\sqrt {1-k^2}+1\right ) \Pi \left (\frac {k^2}{k^2-\sqrt {1-k^2}-1};i \sinh ^{-1}\left (\sqrt {x-1}\right )|\frac {k^2}{k^2-1}\right )-\left (\sqrt {1-k^2}-1\right ) \Pi \left (\frac {k^2}{k^2+\sqrt {1-k^2}-1};i \sinh ^{-1}\left (\sqrt {x-1}\right )|\frac {k^2}{k^2-1}\right )\right )}{k^2 \sqrt {1-k^2} \sqrt {(x-1) x \left (k^2 x-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 189, normalized size = 1.00 \begin {gather*} \frac {i \tan ^{-1}\left (\frac {\sqrt {k^2+2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 x-1}\right )}{2 \sqrt {k^2-1} \sqrt {k^2+2 i \sqrt {k^2-1}-2}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {k^2-2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 x-1}\right )}{2 \sqrt {k^2-1} \sqrt {k^2-2 i \sqrt {k^2-1}-2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 467, normalized size = 2.47 \begin {gather*} \left [\frac {{\left (k^{2} - 1\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) - \sqrt {-k^{2} + 1} \log \left (\frac {k^{4} x^{4} - 4 \, {\left (2 \, k^{4} - k^{2}\right )} x^{3} + 2 \, {\left (4 \, k^{4} + k^{2} - 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 2 \, k^{2} x + 1\right )} \sqrt {-k^{2} + 1} - 4 \, {\left (2 \, k^{2} - 1\right )} x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}, \frac {{\left (k^{2} - 1\right )} \log \left (\frac {k^{4} x^{4} + 4 \, k^{2} x^{3} - 2 \, {\left (3 \, k^{2} + 2\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left (k^{2} x^{2} - 1\right )} + 4 \, x + 1}{k^{4} x^{4} - 4 \, k^{2} x^{3} + 2 \, {\left (k^{2} + 2\right )} x^{2} - 4 \, x + 1}\right ) + 2 \, \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 \, k^{2} x + 1\right )} \sqrt {k^{2} - 1}}{2 \, {\left ({\left (k^{4} - k^{2}\right )} x^{3} - {\left (k^{4} - 1\right )} x^{2} + {\left (k^{2} - 1\right )} x\right )}}\right )}{4 \, {\left (k^{4} - k^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x}{{\left (k^{2} x^{2} - 2 \, x + 1\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 1121, normalized size = 5.93
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x-x^2}{\left (k^2\,x^2-2\,x+1\right )\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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