Optimal. Leaf size=293 \[ \frac {\left (2 i a k^2-2 a \sqrt {k^2-1} k^2-i b k^2-2 b \sqrt {k^2-1}+2 i b\right ) \tan ^{-1}\left (\frac {\sqrt {k^2-2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 (x-1) x}\right )}{2 k^2 \sqrt {k^2-1} \sqrt {k^2-2 i \sqrt {k^2-1}-2}}+\frac {\left (-2 i a k^2-2 a \sqrt {k^2-1} k^2+i b k^2-2 b \sqrt {k^2-1}-2 i b\right ) \tan ^{-1}\left (\frac {\sqrt {k^2+2 i \sqrt {k^2-1}-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 (x-1) x}\right )}{2 k^2 \sqrt {k^2-1} \sqrt {k^2+2 i \sqrt {k^2-1}-2}} \]
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Rubi [C] time = 3.76, antiderivative size = 385, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 10, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6718, 6688, 6728, 6, 714, 115, 934, 12, 168, 537} \begin {gather*} -\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (-2 \sqrt {1-k^2} \left (a k^2+b\right )+2 a k^2+b \left (2-k^2\right )\right ) \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} \left (1-\sqrt {1-k^2}\right ) \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {(1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \left (2 \sqrt {1-k^2} \left (a k^2+b\right )+2 a k^2+b \left (2-k^2\right )\right ) \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} \left (\sqrt {1-k^2}+1\right ) \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (a+\frac {b}{k^2}\right ) F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 6
Rule 12
Rule 115
Rule 168
Rule 537
Rule 714
Rule 934
Rule 6688
Rule 6718
Rule 6728
Rubi steps
\begin {align*} \int \frac {-a-b x+\left (b+a k^2\right ) x^2}{\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 {-a-b x+\left (b+a k^2\right ) x^2}{\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 {-a-b x+\left (b+a k^2\right ) x^2}{\sqrt {1-k^2 x} \sqrt {x-x^2} \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 {a}{\sqrt {1-k^2 x} \sqrt {x-x^2}}+\frac {b}{k^2 \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {b+2 a k^2-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{k^2 \sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+k^2 x^2\right )}\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 {a+\frac {b}{k^2}}{\sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {b+2 a k^2-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{k^2 \sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+k^2 x^2\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\left (a+\frac {b}{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}{\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 {b+2 a k^2-\left (2 a k^2+b \left (2-k^2\right )\right ) x}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (1-2 x+k^2 x^2\right )} \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\left (a+\frac {b}{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}{\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 {-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )}{\sqrt {1-k^2 x} \left (-2-2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}}+\frac {-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )}{\sqrt {1-k^2 x} \left (-2+2 \sqrt {1-k^2}+2 k^2 x\right ) \sqrt {x-x^2}}\right ) \, dx}{k^2 \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \left (a+\frac {b}{k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \left (a+\frac {b}{k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {2} \left (-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\sqrt {2} \left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \left (a+\frac {b}{k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \left (a+\frac {b}{k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (-2 a k^2-b \left (2-k^2\right )-2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \left (-2 a k^2-b \left (2-k^2\right )+2 \sqrt {1-k^2} \left (b+a k^2\right )\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-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \left (a+\frac {b}{k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (b \left (2-k^2-2 \sqrt {1-k^2}\right )+2 a k^2 \left (1-\sqrt {1-k^2}\right )\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} \left (1-\sqrt {1-k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (2 a k^2 \left (1+\sqrt {1-k^2}\right )+b \left (2-k^2+2 \sqrt {1-k^2}\right )\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} \left (1+\sqrt {1-k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ \end {align*}
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Mathematica [C] time = 2.81, size = 244, normalized size = 0.83 \begin {gather*} \frac {i \sqrt {\frac {1}{x-1}+1} (x-1)^{3/2} \sqrt {\frac {1-\frac {1}{k^2}}{x-1}+1} \left (\left (b \left (\sqrt {1-k^2}-1\right )-2 a k^2\right ) \Pi \left (\frac {k^2-1}{k^2-\sqrt {1-k^2}-1};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (2 a k^2+b \sqrt {1-k^2}+b\right ) \Pi \left (\frac {k^2-1}{k^2+\sqrt {1-k^2}-1};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+2 a \sqrt {1-k^2} k^2 F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\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 = 2.73, size = 293, normalized size = 1.00 \begin {gather*} \frac {\left (2 i a k^2-2 a \sqrt {k^2-1} k^2-i b k^2-2 b \sqrt {k^2-1}+2 i b\right ) \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) x}\right )}{2 k^2 \sqrt {k^2-1} \sqrt {k^2-2 i \sqrt {k^2-1}-2}}+\frac {\left (-2 i a k^2-2 a \sqrt {k^2-1} k^2+i b k^2-2 b \sqrt {k^2-1}-2 i b\right ) \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) x}\right )}{2 k^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 = 1.69, size = 491, normalized size = 1.68 \begin {gather*} \left [-\frac {{\left (2 \, a k^{2} + b\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 ) - {\left (b k^{2} - b\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 )}{4 \, {\left (k^{4} - k^{2}\right )}}, \frac {2 \, {\left (2 \, a 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 \, 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 ) + {\left (b k^{2} - b\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 )}{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 {{\left (a k^{2} + b\right )} x^{2} - b x - a}{{\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 = 1907, normalized size = 6.51
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 = -0.00 \begin {gather*} \int -\frac {\left (-a\,k^2-b\right )\,x^2+b\,x+a}{\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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