Optimal. Leaf size=40 \[ -\frac {\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}{x \left (k^2 x^2-1\right )} \]
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Rubi [C] time = 1.71, antiderivative size = 451, normalized size of antiderivative = 11.28, number of steps used = 14, number of rules used = 10, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6719, 21, 6742, 414, 424, 472, 583, 524, 419, 471} \begin {gather*} \frac {k^2 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {k^2 \left (1-x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \left (1-x^2\right ) \left (1-k^2 x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {k \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(k x)|\frac {1}{k^2}\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 k^4 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 414
Rule 419
Rule 424
Rule 471
Rule 472
Rule 524
Rule 583
Rule 6719
Rule 6742
Rubi steps
\begin {align*} \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^2\right )} \, dx &=\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \left (-1+k^2 x^2\right )} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1-2 k^2 x^2+k^2 x^4}{x^2 \sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \left (-\frac {2 k^2}{\sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}}+\frac {1}{x^2 \sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}}+\frac {k^2 x^2}{\sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}}\right ) \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {x^2}{\sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-k^2 x^2\right )^{3/2}} \, dx}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {k^2 \left (1-x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {k^2 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 k^4 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {-1+2 k^2-k^2 x^2}{x^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {1-k^2 x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {-1+k^2 x^2}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {k^2 \left (1-x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {k^2 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 k^4 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \left (1-x^2\right ) \left (1-k^2 x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {k \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(k x)|\frac {1}{k^2}\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {k^2+k^2 \left (1-2 k^2\right ) x^2}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\left (2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {\sqrt {1-k^2 x^2}}{\sqrt {1-x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {k^2 \left (1-x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {k^2 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 k^4 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \left (1-x^2\right ) \left (1-k^2 x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {k \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(k x)|\frac {1}{k^2}\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\left (1-k^2\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2 x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (\left (-1+2 k^2\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2}\right ) \int \frac {\sqrt {1-k^2 x^2}}{\sqrt {1-x^2}} \, dx}{\left (-1+k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ &=\frac {k^2 \left (1-x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {k^2 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {2 k^4 x \left (1-x^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \left (1-x^2\right ) \left (1-k^2 x^2\right )}{\left (1-k^2\right ) x \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {2 k^2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}+\frac {\left (1-2 k^2\right ) \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(x)|k^2\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {k \sqrt {1-x^2} \sqrt {1-k^2 x^2} E\left (\sin ^{-1}(k x)|\frac {1}{k^2}\right )}{\left (1-k^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}-\frac {\sqrt {1-x^2} \sqrt {1-k^2 x^2} F\left (\sin ^{-1}(x)|k^2\right )}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 30, normalized size = 0.75 \begin {gather*} \frac {1-x^2}{x \sqrt {\left (x^2-1\right ) \left (k^2 x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.67, size = 40, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}{x \left (k^2 x^2-1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 36, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}{k^{2} x^{3} - x} \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 {k^{2} x^{4} - 2 \, k^{2} x^{2} + 1}{{\left (k^{2} x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.72 \begin {gather*} -\frac {\left (-1+x \right ) \left (1+x \right )}{\sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}\, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 34, normalized size = 0.85 \begin {gather*} -\frac {x^{2} - 1}{\sqrt {k x + 1} \sqrt {k x - 1} \sqrt {x + 1} \sqrt {x - 1} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 33, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}}{x\,\left (k^2\,x^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{4} - 2 k^{2} x^{2} + 1}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x - 1\right ) \left (k x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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